## Incidence Bounds and Interlacing Eigenvalues

The Szemerédi–Trotter theorem is one of the central results in discrete geometry which gives us a (tight) bound on the number of incidences, i.e., the number of point-line pairs with the point lying on the line, between finite sets of points and lines in the Euclidean plane. More precisely, let $P$ and $L$ be finite sets of points and lines, respectively, in $\mathbb{R}^2$ and define $I(P, L) = \{(p, \ell) \in P \times L \mid p \in \ell\}$. Then we have

$|I(P, L)| \leq C(|P|^{2/3}|L|^{2/3} + |P| + |L|)$,

for some constant $C$. This result can be proved in several ways (including polynomial method), all of which use some topological property of the real plane. The same result does not hold over arbitrary fields, as can be seen for example by taking all $q^2$ points and $q^2 + q$ lines in $\mathbb{F}_q^2$, but some analogous results can be proved over  other fields. The purpose of this post is to discuss one such result and some interesting things around it.

In 2011, Vinh proved that given a set $P$ of points and a set $L$ of lines in $\mathbb{F}_q^2$, we have

$|I(P, L)| \leq \frac{|P||L|}{q} + q^{1/2}\sqrt{|P||L|},$        (1)

using some methods from spectral graph theory. His proof, which is done over the setting of the projective plane $\mathrm{PG}(2, q)$, does not really require any property of these planes over finite fields besides the fact that the points and lines form a combinatorial design (which of course holds for non-Desarguesian planes as well) . This was explicitly observed by Ben Lund and Shubhangi Saraf who proved the following incidence bound on $2$$(v, k, \lambda)$ designs, which generalises the results of Vinh:

Theorem 1. Let $(X, B)$ be a $2$$(v, k, \lambda)$ design with replication number $r$ and number of blocks $b$, let $P$ be a subset of $X$ and let $L$ be a subset of $B$. Then

$||I(P, L)| - \frac{r|P||L|}{b}| \leq (r - \lambda)^{1/2} \sqrt{|P||L|}.$

Since $r = q + 1, b = q^2 + q$ and $\lambda = 1$ for the affine plane $\mathbb{F}_q^2$, we get (1) from this theorem. Moreover, since points and fixed dimensional affine (or projective) subspaces of an $n$-dimensional space over $\mathbb{F}_q$ also form $2$-designs, we get some incidence bounds in those cases as well. The main tool used by Vinh, Lund and Saraf is the following spectral result for bipartite graphs.

Lemma 2. Let $G = (L, R, E)$ be a semiregular bipartite graph with degrees $k_L$ and $k_R$. Let $S \subseteq L$ and $T \subseteq R$ such that $|S| = \alpha|L|$ and $|T| = \beta |R|$. Define $e(S, T) = |\{(x, y) \in S \times T \mid \{x, y\} \in E\}|$. Then we have

$|\frac{e(S,T)}{e(G)} - \alpha \beta| \leq \frac{\lambda_2}{\lambda_1} \sqrt{\alpha \beta(1 - \alpha)(1 - \beta)}$,

where $\lambda_1 = \sqrt{k_Lk_R}$ is the largest eigenvalue of $G$, $\lambda_2$ is the second largest eigenvalue of $G$ in absolute value, and $e(G)$ is the total number of edges in $G$.

Lemma 2 is often referred to as the expander mixing lemma in math and CS literature, and it roughly says that if the second largest eigenvalue of a graph (in absolute value) is small then the graph behaves like a random graph. Though Lemma 2 is usually attributed to a paper of Alon and Chung from 1988 (see for example Section 2.4 of the survey on expander graphs by Hoory, Linial and Wigderson), it actually appeared 9 years before that in the PhD thesis of Willem Haemers (see Theorem 3.1 in this and Theorem 5.1 in this), who proved it using interlacing technique. In fact, Haemers also implicitly proved Theorem 1 by making the observation that the second largest eigenvalue of the incidence graph of a $2$$(v, k, \lambda)$ design is $(r - \lambda)^{1/2}$. Note that for Theorem 1 we simply need to plug in the relevant values in Lemma 2 and ignore the $(1 - \alpha)(1 - \beta)$ term on the right hand side. I’ll sketch the proof of Haemers (see also Section 4.9 in Spectra of Graphs by Brouwer and Haemers). For direct proof of Lemma 2, see Section 3.2 of this paper.

Lemma 3. Let $A$ be a real symmetric matrix with eigenvalues $\lambda_1 \geq \dots \geq \lambda_n$, and let $S$ be an $n \times m$ real matrix satisfying $S^TS = I$. Let $\mu_1 \geq \dots \geq \mu_m$ be the eigenvalues of the (real symmetric) matrix $B = S^T A S$. Then we have $\lambda_i \geq \mu_i \geq \lambda_{n - m + i}$ for all $i \in \{1, \dots, m\}$.

Lemma 3 can be proved using basic properties of Rayleigh quotient. Given two sequences of real numbers $\lambda_1 \geq \dots \geq \lambda_n$ and $\mu_1 \geq \dots \geq \mu_m$, with $m < n$, we say that the second sequence interlaces the first when $\lambda_i \geq \mu_i \geq \lambda_{n - m + i}$ for all $i$ (sometimes the term interlace is used only in the case of $m = n - 1$). Therefore, Lemma 1 tells us that the eigenvalues of the matrix $B$ interlace the eigenvalues of the matrix $A$. By choosing the matrix $S$ appropriately, Lemma 3 has the following corollary for arbitrary graphs.

Corollary 4. Let $G$ be a graph on $n$ vertices and let $X_1, \dots, X_m$ be a partition of its vertices. Define the $m \times m$ quotient matrix $B = (b_{ij})$ of the adjacency matrix $A$ of $G$ with respect to the given partition, by taking $b_{ij}$ to be average row sum of the block of $A$ determined by $X_i$ and $X_j$, i.e., the average number of edges from $X_i$ to $X_j$. Then the eigenvalues of $B$ interlace the eigenvalues of $A$.

Now to prove Lemma 2, we will use the partition $X_1 = S, X_2 = L \setminus S, X_3 = T, X_4 = R \setminus T$ of the bipartite graph $G$. The quotient matrix that we get with respect to this partition is

$B = \begin{bmatrix} 0 & 0 & \frac{e(S,T)}{|S|} & k_L - \frac{e(S,T)}{|S|} \\ 0 & 0 & \frac{k_R|T| - e(S,T)}{|L| - |S|} & k_L -\frac{k_R|T| - e(S,T)}{|L| - |S|} \\\frac{e(S,T)}{|T|} &k_R - \frac{e(S,T)}{|T|} & 0 & 0 \\\frac{k_L|S| - e(S,T)}{|R| - |T|} &k_R -\frac{k_L|T| - e(S,T)}{|R| - |T|} & 0 & 0 \end{bmatrix}$

Now let $\mu_1 \geq \mu_2 \geq \mu_3 \geq \mu_4$ be the eigenvalues of $B$, and let $\lambda_1 \geq \lambda_2 \dots \geq \lambda_{n-1} \geq \lambda_n$ be the eigenvalues of $A$. Then it can be easily shown that $\lambda_1 = \mu_1 = \sqrt{k_Lk_R}$ and $\lambda_n = \mu_4 = -\sqrt{k_Lk_R}$. (Note that the eigenvalues of both $A$ and $B$ are symmetric with respect to the the origin.) From interlacing and the fact that $\mu_3, \lambda_{n-1} < 0$, we get $-\mu_2 \mu_3 \leq -\lambda_2 \lambda_{n-1}$. Since $\det(B)$ is equal to the product of the eigenvalues of $B$, we have

$\det(B)/(k_Lk_R) \leq -\mu_2\mu_3 \leq -\lambda_2 \lambda_{n-1} = \lambda_2^2$.

The determinant of $B$ is not too difficult to compute due to the particular block structure of $B$ and after some simplifications we can see that it is equal to $(k_L k_R)^2 \frac{(e(S,T)/e(G) - \alpha \beta)^2}{\alpha \beta (1 - \alpha)(1 - \beta)}$$\blacksquare$

Corollary 4 and interlacing techniques in general have several other interesting applications too, see the survey by Haemers and the book by Brouwer and Haemers. For another application of Theorem 1 to incidence problems see this paper by Stefaan De Winter, Jeroen Schillewaert and Jacques Verstraete. For a proof of Theorem 1 which does not require spectral graph theory, see this recent paper of Brendan Murphy and Giorgis Petridis. Recently, Ben Lund, Shubhangi Saraf and Charles Wolf have applied Theorem 1 to improve bounds on the finite field Nikodym problem in 3 dimensions, see this. Their work is the first one to show a gap between the lower bound on the size of a Nikodym set and the lower bound on the size of a Kakeya set in $\mathbb{F}_q^3$, for arbitrary $q$.

Edit 05/10/2016: Here is a nice introduction to interlacing techniques and spectral graph theory in general by Haemers: http://upcommons.upc.edu/handle/2099.2/2454. This in particular contains a simple proof of Cheeger’s inequality for regular graphs using interlacing, which I couldn’t find in any other place. More such videos can be found on his webpage.

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## Generalized hexagons containing a subhexagon

I have recently uploaded a joint paper with Bart, “On generalized hexagons of order $(3, t)$ and $(4, t)$ containing a subhexagon”,on arXiv and submitted it for publication. In this work we extend the results of my first paper, which I discussed here, by proving the following:

Let $q \in \{2, 3, 4\}$ and let $S$ be a generalized hexagon isomorphic to the split Cayley hexagon $\mathrm{H}(q)$ or its dual $\mathrm{H}(q)^D$. Then the following holds for any generalized hexagon $S'$ that contains $S$ as a full subgeometry: (1) $S'$ is finite; (2) if $q \in \{2, 4\}$ and $\mathcal{H} \cong \mathrm{H}(q)$, then $S' = S$.

This result is what one might expect in general since (1) we do not know of any semi-finite generalized polygons and (2) we do know of any generalized hexagons which properly contain $\mathrm{H}(q)$ as a full subgeometry, for $q$ a power of a prime $p \neq 3$ (when $q$ is a power of $3$, $\mathrm{H}(q)$ is isomorphic to its dual, which is always contained in a generalized hexagon of order $(q, q^3)$). But since we are nowhere near classifying finite generalized polygons, this problem seems to be pretty hard in general. Also, existence/non-existence of semi-finite generalized polygons is a major open problem in incidence geometry which has only been resolved for specific cases of generalized quadrangles (when every line has $s \in \{3, 4, 5\}$ points on it). So for now we satisfy ourselves with this small contribution.

There is a nice counting based lemma in our paper which I really like and I hope that it can be used to obtain more results for generalized hexagons containing subhexagons. Let’s see what the lemma says.

Let $S$ be a generalized hexagon of order $(s, t)$ contained in a generalized hexagon $S'$ as a full subgeometry (every line of $S$ is also a full line of $S'$). Then using standard arguments for generalized polygons, we first show that every line of $S'$ must have precisely $s + 1$ points on it. It directly follows from the axioms of a generalized hexagon that every point of $S'$ is at distance at most $2$ from $S$. So, we have three “types” of points in $S'$, those contained in $S$, those at distance $1$ from $S$ and those at distance $2$ from $S$. These three types of points can also be characterised by the kind of substructures of $S$ they induce when we take all points of $S$ at non-maximal distance from a given point of $S'$. We call these substructures singular, semi-singular and ovoidal hyperplanes, respectively (there is a good reason why these substructures of $S$ are called “hyperplanes” of $S$; see Section 3 of the paper for exact definitions).

Now if $S$ is a proper subgeometry of $S'$, then it can be shown that there must be a line $L$ of $S'$ which does not contain any point of $S$. Say this line contains $n_L$ points of the third type. Then our lemma says that for any two points $x, y$ on $L$, the substructures of $S$ induced by $x$ and $y$ intersect each other in precisely $s + 1 - n_L$ points. The lemma is proved using a result of Bart (Proposition 4.7 in Polygonal Valuations), which we reprove in the paper without relying too much on the theory of polygonal valuations, followed by simple counting.

How do we use this lemma? Well, if for a given generalized hexagon $S$ one can show that every pair of semi-singular hyperplanes in $S$ intersect each other in more than $s + 1$ points, then there cannot be any generalized hexagon $S'$ be which contains $S$ as a proper full subgeometry as otherwise we can derive a contradiction by finding an appropriate line of $S'$ which does not contain any points of $S$. This is what we check  for the case when $S$ is isomorphic to $\mathrm{H}(2)$ or $\mathrm{H}(4)$ and thus obtain part (2) of our main result. Our check mostly consists of some clever computations in a computer model of these geometries as we do not really understand how semi-singular (or ovoidal) hyperplanes of split Cayley hexagons and their duals behave in general.

To prove part (1) of our main result we show that if every point of $S'$ is at distance at most $1$ from $S$, or equivalently, there are no points of the third type in $S'$, then through every point of $S'$ there are only finitely many lines. Therefore, if we show that a given generalized hexagon $S$ does not contain any ovoidal hyperplanes a.k.a. $1$-ovoids a.k.a. distance-$2$ ovoids, then from this result we get that every generalized hexagon containing $S$ as a full subgeometry is finite. These $1$-ovoids are simply sets of points which intersect every line in a unique point, thus in graph theoretical terms these are the exact hitting sets of the hypergraph obtained from the generalized hexagon by identifying each line by the set of points it contains. Now when $S$ is isomorphic to $\mathrm{H}(2)^D$ (a geometry that has $63$ points and $63$ lines) it is easy to show via an exhaustive computer search or via an old result of Frohardt and Johnson which classifies all hyperplanes of this geometry, that it does not contain any $1$-ovoids. But in general this problem is quite hard. For $\mathrm{H}(4)^D$ we use the non-existence result proved by me and Ferdinand in a recent paper.

The last remaining case of our main result is when $S$ is isomorphic to $H(3)$. The extra complication here is that $H(3)$ is isomorphic to its dual, and hence it is contained in the dual twisted triality hexagon of order $(3, 27)$. Moreover, it has $1$-ovoids. What we do here is again use the lemma on intersection sizes, but this time we show that for every point $x$, every semi-singular hyperplane  with center $x$ (again, see Section 3 of the paper for the definition) of $S$ intersects every ovoidal hyperplane containing $x$ in more than $3$ points. From this we show that there are no type 3 points in any generalized hexagon $S'$ which contains $S$ has a full subgeometry.

Sadly, we have reached our computational limitations at $q = 4$ and it seems like for larger cases we will need more ideas and a better understanding of these three types of hyperplanes in split Cayley hexagons and their duals. If in general one can prove that every semi-singular hyperplane of $\mathrm{H}(q)$  (or $\mathrm{H}(q)^D)$ through a fixed point $x$  intersects every ovoidal hyperplane containing $x$ (whenever such a hyperplane exists) in more than $q$ points, then this would imply that there is no semi-finite generalized hexagon with $q + 1$ points on each line, containing $\mathrm{H}(q)$ (or $\mathrm{H}(q)^D)$ as a subgeometry. That would be really nice.

There are some natural open problems arising from this work which we list in the last section of our paper. In particular, I would be really happy to see a proof of the following conjecture in the near future.

Every generalized hexagon containing the generalized hexagon of order $(2, 1)$, which arises from the incidence graph of the Fano plane, is finite.

This is the smallest case where none of our techniques work.

## Applications of Alon-Furedi to finite geometry

In a previous post I discussed how the Alon-Furedi theorem serves as a common generalisation of the results of Schwartz, DeMillo, Lipton and Zippel. Here I will show some nice applications of this theorem to finite geometry (reference: Section 6 of my paper with Clark, Potukuchi and Schmitt). First, let’s recall the statement of Alon-Furedi.

Theorem 1 (Alon-Furedi). Let $A = A_1 \times \cdots \times A_n$ be a finite grid in $\mathbb{F}^n$ where $|A_i| = a_i$. Let $f \in \mathbb{F}[t_1, \dots, t_n]$ be an $n$-variable polynomial of degree $d$ such that the set $\mathcal{U}_A(f) = \{x \in A : f(x) \neq 0\}$ is nonempty (in other words, $f$ does not vanish on the entire grid). Then $|\mathcal{U}_A(f)| \geq \mathfrak{m}(a_1, \dots, a_n; \sum_{i = 1}^n a_i - d)$. Moreover, the bound is sharp.

Here $\mathfrak{m}(a_1, \dots, a_n; k)$ denotes the minimum value of the product $y_1 \cdots y_n$ where $y_i$‘s are integers satisfying $y_1 + \cdots + y_n = k$ and $1 \leq y_i \leq a_i$ for all $i$. See my post on balls in bins for some details about this function. Thus, the Alon-Furedi theorem gives us a sharp lower bound on the number of non-zeros a polynomial function of degree at most $d$ in a finite grid, given that the polynomial does not vanish on the entire grid.

If we take $\mathbb{F}$ to be the finite field $\mathbb{F}_q$, and $A_i = \mathbb{F}_q$ for all $i$, then this theorem is in fact equivalent to a 1968 result of Kasami, Lin and Peterson, that determines the minimum Hamming distance of generalized Reed-Muller codes. The generalized Reed-Muller code over $\mathbb{F}_q$ is obtained by evaluating each reduced polynomial (degree in each variable at most $q - 1$, see this) of degree at most $d$ on the points of $\mathbb{F}_q^n$. Clearly, the minimum Hamming distance of this linear code is the minimum number of non-zeros a reduced polynomial of degree at most $d$ can have. But that’s precisely what the Alon-Furedi theorem is about! After an easy computation one can show that if $d = a(q - 1) + b$ where $0 < b \leq q - 1$, then this minimum is equal to $(q - b)q^{n - a - 1}$, which is what Kasami-Lin-Peterson proved in 1968 (using more technical arguments involving BCH codes). This particular case of Alon-Furedi is what we need for most of our applications to finite geometry, and thus these results can also be seen as applications of the Kasami-Lin-Peterson theorem.

First let’s show that the original problem studied by Alon-Furedi, which also appeared as problem 6 in 2007 International Maths Olympiad, can be solved using Theorem 1. We are given a finite grid $A_1 \times \cdots \times A_n$  in $\mathbb{F}^n$ and we are asked to show that the number of hyperplanes required to cover all points of the grid except one is equal to $\sum (|A_i| - 1)$. Associate each hyperplane to the linear polynomial that defines it, and take the product of these linear polynomials. Then the set of points covered by the hyperplanes is equal to the set of zeros of this polynomial. If the degree of this polynomial, which is equal to the number of hyperplanes, is less than $\sum (|A_i| - 1)$, then by Theorem 1 there are at least $\mathfrak{m}(|A_1|, \dots, |A_n|; n + 1) \geq 2$ points of the grid which are not covered, a contradiction.

Now let’s look at this problem of hyperplane covering for the finite projective and affine spaces over $\mathbb{F}_q$. It would be important to remember that the projective space $PG(n, q)$ can be obtained from the affine space $AG(n, q)$ (which is isomorphic to $\mathbb{F}_q^n$ after fixing a point), by adding a hyperplane at infinity. Let $H_1, \dots, H_k$ be $k$ hyperplanes in $PG(n, q)$ which do not cover all the points. Treating $H_k$ as the hyperplane at infinity, we get hyperplanes $H_1', \dots, H_{k-1}'$ in $\mathbb{F}_q^n$ which do not cover all the points of the grid $\mathbb{F}_q^n$. Therefore, by Theorem 1, there are at least $\mathfrak{m}(q, \dots, q; nq - k + 1)$ points of $PG(n, q)$ which are missed by these hyperplanes. Such a collection of hyperplanes is known as a partial cover and the points missed are known as holes. By computing this function, we have the following corollary, which is a stronger version of a result of S. Dodunekov, L. Storme and G. Van de Voorde:

If $0 \leq a < q$, then a partial cover of $PG(n, q)$ of size $q + a$ has at least $q^{n-1} - aq^{n-2}$ holes.

By projective duality, we can translate our statement to sets of points in $PG(n, q)$ which do not meet all hyperplanes. In fact, we get the following nice result for affine spaces by noticing that a subset of affine points in $PG(n, q)$ does not meet the hyperplane at infinity:

Theorem 2. Let $S$ be a set of $k$ points in $AG(n, q)$. Then there are at least $\mathfrak{m}(q, \dots, q; nq - k + 1) - 1$ hyperplanes of $AG(n, q)$ which do not meet $S$.

A direct corollary of Theorem 2 is the famous result of Jamison/Brouwer-Schrijver on affine blocking sets (sets of points which meet all hyperplanes, i.e., contain at least one point of each hyperplane):

The minimum number of points required to block all hyperplanes in $AG(n, q)$ is $n(q - 1) + 1$.

(Proof: if you take less than that many points, then by Theorem 2, there will be at least one hyperplane which does not meet the set of points)

We can obtain another interesting result on blocking sets, this time in $PG(n, q)$. A point $x$ of a blocking set $B$ in $PG(n, q)$ is called an essential point if removing $x$ from $B$ we have a set which does not meet all hyperplanes. Clearly, through every essential point of $B$, there must be a hyperplane $H$ such that $H \cap B = \{x\}$. These are known as tangent hyperplanes. How many tangent hyperplanes can be there through an essential point of a blocking set?

Theorem 3. Let $B$ be a blocking set in $PG(n, q)$ and $x$ an essential point of $B$. Then there are at least $\mathfrak{m}(q, \dots, q; nq - |B| + 2)$ tangent hyperplanes through $x$.

(Proof: fix a hyperplane through $x$ and treat it as the hyperplane at infinity; removing this hyperplane reduces the problem to Theorem 2 if we take $S = B \setminus \{x\}$.)

A nice direct corollary of Theorem 3 which I noticed today is a lower bound on the size of a blocking semioval that is almost the same as the bound obtained by Dover in 2000. A semioval in $PG(2, q)$ is a set $S$ of points with the property that for every point $x$ in $S$, there is a unique line tangent to $S$ at $x$ (this property is satisfied by ovals but it does not characterise them). The classical examples of semiovals are obtained from polarities of projective planes. A semioval is called a blocking semioval if it is also a blocking set. There is a general lower bound of $q + \sqrt{q} + 1$ on the size of a blocking set in an arbitrary projective plane of order $q$ given by Bruen, but for blocking semiovals we can obtain a much better lower bound. From Theorem 3, it follows that if $|B| < 2q$, then there are at least $\mathfrak{m}(q, q; 3) \geq 2$ tangent lines through an essential point of $B$. Therefore, if $B$ is a blocking set of size less than $2q$, then $B$ cannot be a semioval.

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## The Ellenberg-Gijswijt bound on cap sets

Four days back Jordan Ellenberg posted the following on his blog:

Briefly:  it seems to me that the idea of the Croot-Lev-Pach paper I posted about yesterday can indeed be used to give a new bound on the size of subsets of F_3^n with no three-term arithmetic progression! Such a set has size at most (2.756)^n. (There’s actually a closed form for the constant, I think, but I haven’t written it down yet.)

Here’s the preprint. It’s very short. I’ll post this to the arXiv in a day or two, assuming I (or you) don’t find anything wrong with it, so comment if you have comments!

Then two days later Dion Gijswijt made this comment on Ellenberg’s post:

What a nice result, congratulations!

Last week, I’ve also been writing a proof of an upper bound O(p^{cn}), c<1 for progression-free sets, in Z_p^n based on the same paper of Croot, Lev and Pach, see

http://homepage.tudelft.nl/64a8q/progressions.pdf

So I guess, you beat me to it! Also, your constant for the case p=3 is better than mine (O(2.76^n) vs O(2.84^n)). I think the CLP-paper is a real gem in its simplicity.

This is a true breakthrough, settling an important open problem in combinatorics that survived even after a lot of effort from several mathematicians (see this, this, this and this). Ultimately, just like the finite field Kakeya problem, it succumbed to the polynomial method. I really like the following comment by Ben Green on a Facebook post: “It’s amazing how the polynomial method seems to be so orthogonal to other methods. It’s a bit like Dvir’s proof of the finite field Kakeya – in a couple of pages, it sweeps aside a large body of partial results using complicated combinatorial and analytic methods to get much weaker statements.”

Problem statement: Find the maximum possible size of a subset of $\mathbb{F}_q^n$ which does not contain any three distinct elements in arithmetic progression.

A set with such a property is known as a cap set, a name probably inspired from the fact that the early examples of such sets in three dimension came from elliptic quadrics over finite fields, which look like hats (or caps) in real spaces. The elliptic quadrics are 3-dimensional examples of affine caps; sets of points  which do not contain any three points collinear with the same line (this condition is stronger than the condition above when $q > 3$ as it says that for any three distinct elements $a, b, c$ we have $c - b \neq \lambda (b -a)$ for all $\lambda \in \mathbb{F}_q \setminus \{0, -1\}$.). The oldest reference that I could find mentioning the term “cap” for such sets is this 1958 paper by Segre: On Galois Geometries. When $q = 3$, affine caps are related to the game SET and you can read the following brilliant paper by B. L. Davis and D. Maclagan which explains this connnection: The card game SET.

Ellenberg and Gijswijt  have proved that a cap set in $\mathbb{F}_q^n$ has size at most a constant times the number of monomials $x_1^{e_1}x_2^{e_2}\cdots x_n^{e_n}$ of total degree at most $(q-1)n/3$ that satisfy $0 \leq e_i \leq q - 1$ for all $i$.

Understanding their beautiful proof requires only some knowledge of basic linear algebra and polynomials. I’ll try to explain the main arguments here. Ultimately, one also needs to estimate the total number of monomials that satisfy the conditions above, for which I will refer to the preprints of Ellenberg and Gijswijt, and to the comments section of the blog post by Ellenberg. One crucial thing to note about the estimates is that this bound is good only because $(q-1)n/3$ is strictly less than the mean $(q-1)n/2$.

Let $q$ be an odd prime power and let $\mathcal{P}_d(n, q)$ denote the set of all polynomials $f$ in $\mathbb{F}_q[x_1, \dots, x_n]$ that satisfy $\deg f \leq d$ and $\deg_{x_i} f \leq q - 1$ for all $i$, aka, the set of reduced polynomials of degree at most $d$. The set of all reduced polynomials with no restriction on the total degree is simply denoted by $\mathcal{P}(n, q)$. There is a vector space isomorphism between $\mathcal{P}(n, q)$ and the space of all $\mathbb{F}_q$-valued functions on $\mathbb{F}_q^n$ given by evaluating the polynomials. Let $k_d$ denote the dimension of $\mathcal{P}_d(n, q)$, which is equal to the total number of integral solutions of $e_1 + \dots + e_n \leq d$ and $0 \leq e_i \leq q - 1$. Clearly, $k_{n(q - 1) - d} = q^n - k_d$ via the bijection $(e_1, \dots, e_n) \mapsto (q - 1 - e_1, \dots, q - 1 - e_n)$.

For a $3$-term arithmetic progression free subset $A$ of $\mathbb{F}_q^n$, the sets $A + A = \{a + a' : a, a' \in A, a \neq a'\}$ and $2A = \{a + a : a \in A\}$ are disjoint as otherwise we will have three distinct elements $a, b, c$ in $A$ satisfying $a + c = 2b$.
Let $U$ be the subspace of $\mathcal{P}(n, q)$ consisting of all polynomials that vanish on the complement of $2A$.
Then $\dim U = |2A| = |A|$ (since $q$ is odd) and thus we try to find upper bounds on $\dim U$.
For any integer $d \in \{0, 1, \dots, n(q - 1)\}$ let $U_d$ be the intersection of $\mathcal{P}_d(n, q)$ with $U$.
Then since the span $\langle U, \mathcal{P}_d(n, q) \rangle$ is a subspace of  $\mathcal{P}(n, q)$, by the dimension formula we have $\dim U \leq \dim \mathcal{P}(n, q) - \dim \mathcal{P}_d(n, q) + \dim U_d$, or equivalently, $|A| \leq q^n - k_d + \dim U_d = k_{n(q - 1) - d} + \dim U_d.$

Now the crux of the proof is Proposition 1 in Jordan’s preprint, which is essentially equivalent to Lemma 1 of Croot-Lev-Pach’s paper, and says that every (reduced) polynomial of degree at most $d$ which vanishes on $A + A$ has at most $2 k_{d/2}$ non-zeros in $2A$. The proof of this is an interesting dimension argument which I will skip for now (also see this old mathoverflow question by Seva).

Since $A + A$ is a subset of the complement of $2A$, the above proposition implies that every element of $U_d$, when seen as an element of $\mathbb{F}_q^{\mathbb{F}_q^n}$ via the evaluation isomorphism, has at most $2 k_{d/2}$ non-zero coordinates. I leave it as an exercise to show that a vector subspace $V$ of $F^n$ with the property that every element of $V$ has at most $k$ non-zero coordinates must have its dimension less than or equal to $k$ (hint: row reduction). And thus $\dim U_d \leq 2 k_{d/2}$.

Therefore, from the earlier inequality on $|A|$, we get $|A| \leq k_{n(q - 1) - d} + 2k_{d/2}$.
Finally, observe that $n(q - 1) - d = d/2$ when $d = 2(q - 1)n/3$, which gives us the bound $|A| \leq 3 k_{(q - 1)n/3}$ (whenever $(q - 1)n/3$ is an integer).
The reason why this is a good bound is that $k_{d}$ is bounded above by $q^{\lambda n}$ for some $\lambda < 1$ whenever $d < (q - 1)n/2$ (see the preprints).

Some questions:
(1) Can we obtain a better bound on affine caps since they are much more restrictive than cap sets?
(2) Do these techniques say anything about sets which do not contain any full line, i.e., the complements of affine blocking sets with respect to lines? As Ferdinand has pointed out in this comment, we have a general upper bound of the form $q^n - 2q^{n - 1} + 1$ there.
(3) What about sets which do not contain any $k$ terms in arithmetic progression, for $k > 3$?

Further Reading:
[1] Mind Boggling: Following the work of Croot, Lev, and Pach, Jordan Ellenberg settled the cap set problem! by Gil Kalai
[2] Polymath 10 Emergency Post 5: The Erdos-Szemeredi Sunflower Conjecture is Now Proven. by Gil Kalai
[3] A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound by Terence Tao
[4] Many Zero Divisors in a Group Ring Imply Bounds on Progression–Free Subsets by Fedor Petrov
[5] Large caps in projective Galois spaces, by Jürgen Bierbrauer and Yves Edel
[6] Zero-sum problems in finite abelian groups and affine caps, by Edel et al.
[7] Sumsets and sumsets of subsets, by Jordan Ellenberg.
[8] Bounds for sum free sets in prime power cyclic groups – three ways, by David speyer.
[9] Talk by Jordan Ellenberg at Discrete Math Seminar in IAS, Princeton.

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## The Erdős-Ginzburg-Ziv theorem

Let $S = (a_1, \dots, a_n)$ be  a sequence of integers (not necessarily distinct). Then there exists a subsequence of $S$ the sum of whose elements is divisible by $n$

This is one of the first problems I saw when learning the pigeonhole principle. And it’s a good exercise to try it yourself by taking the pigeonhole principle as a hint. For our purpose, it’s best stated and proved in the following form.

Theorem 1. Let $S = (g_1, \dots, g_n)$ be a sequence of elements of an abelian group $(G, +, 0)$ of order $n$. Then there exists a non-empty subsequence of $S$ that sums up to $0$

Proof. Define $s_1 = g_1$, $s_2 = g_1 + g_2$, $\dots$, $s_n = g_1 + \dots + g_n$. Say none of the $s_i$‘s are zero. Since there are $n - 1$ non-zero elements in $G$, we must have $s_i = s_j$ for some $i < j$ by the pigeonhole principle. Hence $s_j - s_i = \sum_{k = i+1}^j a_k = 0$.

In 1961, Erdős, Ginzburg and Ziv proved an interesting similar looking result which later became one of the pioneering results in the field of additive combinatorics/combinatorial number theory/arithmetic combinatorics. It says that given a sequence of $2n - 1$ integers, $n$ of them must have their sum divisible by $n$. In fact, the original paper  implicitly proves the following more general result.

Theorem 2 (EGZ). Let $S = (g_1, \dots, g_{2n-1})$ be a sequence of elements of an abelian group $(G, +, 0)$ of order $n$. Then there exists a non-empty subsequence of $S$ that sums up to $0$ and has size $n$

There is an induction step in the paper which shows that it suffices to prove Theorem 2 in the special case when $G$ is the cyclic group of prime order $p$, i.e., $G \cong (\mathbb{F}_p, + , 0)$ where $\mathbb{F}_p$ is the finite field of order $p$. In this post I will discuss the proof of this prime case using the Alon-Furedi theorem, which was originally discovered by Clark, Forrow and Schmitt in Warning’s Second Theorem with Restricted Variables and then rediscovered by me and potu. The proof is quite similar to the 1989 proof by C. Bailey and R.B. Richter which used the famous Chevalley-Warning theorem, but instead we use a generalisation of CW; the so-called Restricted-Variable Chevalley-Warning theorem due to David Brink that follows directly from Alon-Furedi. I believe that this is a more natural proof than the one that uses Chevalley-Warning.

Theorem 3 (RV CW). Let $P_1, \dots, P_r \in \mathbb{F}_q[t_1, \dots, t_n]$. For $1 \leq i \leq n$, let $A_i \subseteq \mathbb{F}_q$ and put $A = \prod_{i = 1}^n A_i \subseteq \mathbb{F}_q^n$. If $\sum (q -1) \deg P_i < \sum (|A_i| - 1)$, then $P_i$‘s can’t have a unique common zero in $A$

Proof. Let $P = \prod (1 - P_i^{q-1})$. Say all $P_i$‘s have a common zero $a \in A$. Then $P(a) \neq 0$, so $P$ is a non-zero polynomial of degree $d = \sum (q - 1)\deg P_i < \sum (|A_i| - 1)$. Thus, by Alon-Furedi it has at least $\mathfrak{m}(|A_1|, \dots, |A_n|; \sum |A_i| - d) \geq 2$ non-zeros in $A$, which gives us at least two common zeros of $P_i$‘s.

We are now ready to prove the EGZ theorem for $G \cong \mathbb{F}_p$. Let $a_1, \ldots, a_{2p - 1}$ be elements of $\mathbb{F}_p$. Let $f = \sum a_i t_i$ and $g = \sum t_i$ be two polynomials in $\mathbb{F}_p[t_1, \dots, t_{2p-1}]$. Then $(0, \dots, 0)$ is a common zero of $f$ and $g$ and since $\deg f + \deg g = 2p - 2 < 2p-1$, there exists another common zero $x = (x_1, \dots, x_{2p-1})$ in $\{0, 1\}^{2p-1}$. Let $i_1, \dots, i_k$, be all the positions where $x_i$ is equal to $1$. Then $f(x) = 0$ gives us $a_{i_1} + \dots + a_{i_k} = 0$ and $g(x) = 0$ gives us $k \equiv 0 \pmod{p}$. Since $0 < k < 2p$, we must have $k = p$. $\blacksquare$

For the sake of completeness, let’s give the induction argument also which will completely prove Theorem 2. We say that a finite abelian group $(G, +, 0)$ satisfies the EGZ property if any sequence of $2|G| - 1$ elements of $G$ contains a subsequence of length $|G|$ that sums up to $0$.

Lemma 4. Let $G$ be a finite abelian group and $N$ a subgroup of $G$If both $N$ and $G/N$ satisfy the EGZ property, then $G$ also satisfies the EGZ property.

Proof. Say $|N| = n$ and $|G/N| = r$. Let $S = (g_1, \dots, g_{2nr - 1})$ be a sequence of elements of $G$.
Since $G/N$ satisfies the EGZ property, from the elements $g_1 + N, \dots, g_{2r - 1} + N$ of $G/N$ we can choose $r$ which sum to the identity of $G/N$. By renaming the elements say $g_1 + N, \dots, g_r + N$ are those elements, i.e., $(g_1 + \dots + g_r) + N = N$, or equivalently we have $h_1 = g_1 + \dots + g_r \in N$. Now look at the first $2r - 1$ elements of the remaining $2nr - 1 - r$ elements of $S$ and repeat the argument to get $h_2 = g_{r+1} + \dots + g_{2r} \in N$. We can repeat this procedure till we have $h_{2n - 1} = g_{(2n - 2)r + 1} + \dots + g_{(2n - 1)r} \in N$.
Since $N$ satisfies the EGZ property, after renaming $h_i$‘s, we have $h_1 + \dots + h_n = 0$. By plugging in the values of $h_1, \dots, h_n$ in terms of the elements of $S$ we obtain $nr$ elements of $G$ that sum up to $0$.

Lemma 4 along with the fundamental theorem of finite abelian groups now proves the theorem. In fact, one can prove a similar lemma for non-abelian groups and use it to prove EGZ for all solvable groups, as was done by B. Sury in this paper. For more discussion on this result, see these notes by Pete or any standard textbook on additive combinatorics/combinatorial number theory. Also see this famous paper by Alon and Dubiner that discusses several proofs of this result.

Posted in Combinatorics, Polynomial Method | | 8 Comments

## Alon-Furedi, Schwartz-Zippel, DeMillo-Lipton and their common generalization

In the post Balls in Bins I wrote about a combinatorial function $\mathfrak{m}(a_1, \dots, a_n; k)$ which denotes the minimum value of the product $y_1 \cdot y_2 \cdots y_n$ among all distributions $(y_1, y_2, \dots, y_n)$ of $k$ balls (so $y_1 + y_2 + \dots + y_n = k$) in $n$ bins with the constraints $1 \leq y_i \leq a_i$. It turns out that this combinatorial function is linked to zeros of a multivariable polynomial in a “finite grid”. Here, by a finite grid I mean a subset $A \subseteq \mathbb{F}^n$ of the form $A = A_1 \times \cdots \times A_n$ where $\mathbb{F}$ is a field and $A_i$‘s are finite subsets of $\mathbb{F}$. The Alon-Furedi theorem, which provides this link, is as follows:

Theorem 1 (Alon-Furedi). Let $A = A_1 \times \cdots \times A_n$ be a finite grid in $\mathbb{F}^n$ where $|A_i| = a_i$. Let $f \in \mathbb{F}[t_1, \dots, t_n]$ be an $n$-variable polynomial of degree $d$ such that the set $\mathcal{U}_A(f) = \{x \in A : f(x) \neq 0\}$ is nonempty (in other words, $f$ does not vanish on the entire grid). Then $|\mathcal{U}_A(f)| \geq \mathfrak{m}(a_1, \dots, a_n; \sum_{i = 1}^n a_i - d)$. Moreover, the bound is sharp.

For the sharpness of this bound let $(y_1, \dots, y_n)$ be any feasible distribution of $\sum_{i = 1}^n a_i - d$ balls in bins. Pick subsets $B_i \subseteq A_i$ with $|B_i| = y_i$ and define $f = \prod_{i = 1}^n \prod_{\lambda \in A_i \setminus B_i} (t_i - \lambda)$ which has degree $\sum_{i = 1}^n (a_i - y_i) = d$ and doesn’t vanish on $y_1 \cdot y_2 \cdots y_n$ points of the grid $A$.

This theorem appears in the last pages of [1] as Theorem 5, and until the appearance of [4] where Clark, Forrow and Schmitt used it to obtain a generalization of Warning’s second theorem and several interesting results in combinatorics and additive number theory, it seems like this result was largely ignored. In [3] Pete further generalized the results of [4] and gave some nice applications to graph theory, additive group theory and polynomial interpolation. In fact, he also generalized the Alon-Füredi theorem by replacing the field $\mathbb{F}$ by a commutative ring $R$ and imposing some extra conditions on the grid which are vacuously true for the field case. While discussing some results of [4] with Pete and John I realised that we can do much more with this Alon-Furedi theorem, and this resulted in my joint paper with Pete, Aditya and John [2], which I have also discussed here. A particular contribution of [2] is the following generalization of Alon-Furedi (Theorem 3 below) for which we first need some motivation.

The famous Schwartz-Zippel lemma says that a polynomial in $\mathbb{F}_q[t_1, \dots, t_n]$ of degree $d \leq q$ has at most $dq^{n-1}$ zeros in $\mathbb{F}_q^n$. In fact it works over more general grids as well where it says that if $A_1, A_2, \dots, A_n$ are finite subsets of $\mathbb{F}$ with $|A_1| \geq |A_2| \geq \cdots \geq |A_n|$ and $f$ a polynomial of degree $d \leq |A_n|$, then $f$ has at most $d|A_1||A_2|\cdots|A_{n-1}|$ zeros in $A_1 \times \dots \times A_n$. Since Theorem 1 gives a lower bound on the number of non-zeros of a polynomial in a finite grid, it automatically gives an upper bound on the numbers of zeros of that polynomial. This observation leads to a natural question.

When the degree of the polynomial is at most $|A_n|$, does Theorem 1 imply the Schwartz-Zippel lemma?

If you do the math, then you would see that indeed, it does. So, the Alon-Füredi theorem is in fact a generalization of the so-called Schwartz-Zippel lemma (see the blog post by RJ Lipton and some comments on it for why I have used “so-called” in this sentence). But, there are other results in the spirit of Schwartz-Zippel lemma, which do not follow from Theorem 1! The following result is due to DeMillo-Lipton and Zippel:

Theorem 2. Let $f \in \mathbb{F}[t_1, \dots, t_n]$ be a non-zero polynomial such that for all $i$, we have $\deg_{t_i} f \leq d$ for some constant $d$. Then for a subset $S \subseteq \mathbb{F}$ of cardinality at least $d + 1$, the number of zeros of $f$ in $S^n$ is at most $|S|^n - (|S| - d)^n$

To see that this result doesn’t follow from Theorem 1, take $S = \{0, 1, 2\} \subseteq \mathbb{Q}$ and $f = t_1 t_1 \in \mathbb{Q}[t_1, t_2]$. Then Theorem 1 tells us that $f$ has at most $6$ zeros in $S^2$, while Theorem 2 tells us that $f$ has at most $5$ zeros (which is stronger and the sharp bound in this case). This doesn’t mean that DeMillo-Lipton/Zippel’s result is always better. In fact, if you take $f = t_1 + t_2$, then Theorem 1 gives the better bound. To “rectify” this situation was one of our main motivation to give the following generalization of Theorem 1, which appears as Theorem 1.2 in [2].

Theorem 3 (Generalized Alon-Furedi). Let $A = A_1 \times \cdots \times A_n$ be a finite grid in $\mathbb{F}^n$ where $|A_i| = a_i$.  For $i \in \{1, 2, \dots, n\}$ let $b_i$ be an integer such that $1 \leq b_i \leq a_i$. Let $f \in \mathbb{F}[t_1, \dots, t_n]$ be an $n$-variable polynomial of degree $d$ such that the set $\mathcal{U}_A(f) = \{x \in A : f(x) \neq 0\}$ is nonempty and $\deg_{t_i} f \leq a_i - b_i$ for all $i$. Then we have $|\mathcal{U}_A(f)| \geq \mathfrak{m}(a_1, \dots, a_n; b_1, \dots, b_n; \sum_{i = 1}^n a_i - d)$. Moreover, the bound is sharp.

Here $\mathfrak{m}(a_1, \ldots, a_n; b_1, \ldots, b_n; k)$ is defined to be the minimum value of the product $y_1 \cdot y_2 \cdots y_n$ among all distributions $(y_1, y_2, \dots, y_n)$ of $k = y_1 + y_2 + \dots + y_n$ balls in $n$ bins with the constraints $b_i \leq y_i \leq a_i$. If you “do the math” (see Section 4 of [2]), then Theorem 3 is a common generalization of Theorem 1, Schwartz-Zippel lemma and Theorem 2.

References
[1] N. Alon and Z. Füredi, Covering the cube by affine hyperplanes. Eur. J. Comb. 14 (1993), 79–83.
[2] A. Bishnoi, P. L. Clark, A. Potukuchi, J. R. Schmitt, On Zeros of a Polynomial in a Finite Grid (2015). arXiv preprint arXiv:1508.06020.
[3] P. L. Clark, Fattening up Warning’s Second Theorem (2015). arXiv preprint arXiv:1506.06743.
[4] P. L. Clark, A. Forrow and J. R. Schmitt, Warning’s Second Theorem With Restricted Variables. To appear in Combinatorica, (2016).

Posted in Combinatorics, Polynomial Method | | 2 Comments

## A timeline of the polynomial method up-to combinatorial nullstellensatz

Over the past 30-40 years, the so-called polynomial method has developed into a powerful tool in combinatorics and (additive) number theory. There has been a lot of recent interest in it after Dvir’s  paper on the Kakeya conjecture, where he solved a major open problem by a short and easy polynomial argument. After that, many prominent mathematicians from different areas of mathematics started looking into these techniques, and in some cases, ended up resolving some famous open problems: On the Erdős distinct distances problem in the plane.

In this post I will give a timeline of some important papers (in my opinion) related to the polynomial method, starting from Tom H. Koornwinder’s paper on equiangular lines till the appearance of Combinatorial Nullstellensatz by Noga Alon (1999) (so, this timeline will cover only the part A of polynomial method as mentioned here). But first, here are some quotes that describe the general philosophy behind the polynomial method:

Given a set of points (or vectors, or sets) that satisfy some property, we want to say something about the size or the structure of this set. The approach is then to associate to this set a polynomial, or a collection of polynomials, and use properties of polynomials to obtain information on the size or structure of the set.

– Aart Blokhuis, 1993

Broadly speaking, the strategy is to capture (or at least partition) the arbitrary sets of objects (viewed as points in some configuration space) in the zero set of a polynomial whose degree (or other measure of complexity) is under control; for instance, the degree may be bounded by some function of the number of objects. One then uses tools from algebraic geometry to understand the structure of this zero set, and thence to control the original sets of object.

– Terence Tao, 2013

1975
A note on the absolute bound for systems of lines, by Koornwinder

1977
On Two-Distance Sets in Euclidean Space, by Larmen, Rogers and Seidel
Covering finite fields with cosets of subspaces, by Jamison

1978
The blocking number of an affine space, by Brouwer and Schrijver

1981
A New Upper Bound for The Cardinality of 2-Distance Sets in Euclidean Space, by Blokhuis
Intersection theorems  with geometric consequences, by Frankl and Wilson
Remarks on a theorem of Rédei, by Lovasz and Schrijver

1984
Regular subgraphs of almost regular graphs, by Alon, Friedland and Kalai
Every 4-regular graph plus an edge contains a 3-regular subgraph, by Alon, Friedland and Kalai

1987
– A characterization of exterior lines of certain sets of points in PG (2, q), by Blokhuis and Wilbrink

1988
A nowhere zero point in linear mappings, by Alon and Tarsi
A short proof of the nonuniform Ray-Chaudhuri-Wilson inequality, by Babai
Balancing set of vectors, by Alon, Bergmann, Coppersmith and Odlyzko

1989
Sum zero (mod n), size n subsets of integers, by Bailey and Richter

1991
Multilinear polynomials and Frankl-Ray-Chaudhuri-Wilson type intersection theorems, by Alon, Babai and Suzuki

1992
Polynomial multiplicities over finite fields and intersection sets, by Bruen
Colorings and orientations of graphs, by Alon and Tarsi

1993
The Dinitz problem solved for rectangles, by Janssen
Covering the Cube by Affine Hyperplanes, by Alon and Furedi
Polynomials in finite geometries and combinatorics, by Blokhuis
Zero-sum sets of prescribed size, by Alon and Dubiner

1994
On the size of a blocking set in PG(2,p), by Blokhuis
On nuclei and affine blocking sets, by Blokhuis
– On multiple nuclei and a conjecture of Lunelli and Sce, by Blokhuis

1995
Tools from higher algebra, by Alon
Adding Distinct Congruence Classes Modulo a Prime, by Alon, Nathanson and Ruzsa
A new proof of several inequalities on codes and sets, by Babai, Snevily and Wilson
The number of directions determined by a function f on a finite field, by Blokhuis, Brouwer and Szonyi

1996
The Polynomial Method and Restricted Sums of Congruence Classes, by Alon, Nathanson and Ruzsa
Multiple blocking sets and arcs in finite planes, by Ball

1998
An easier proof of the maximal arcs conjecture, by Ball and Blokhuis
Sumsets in Vector Spaces over Finite Fields, by Eliahou and Kervaire

1999
– Multilinear Polynomials and a Conjecture of Frankl and Füredi, by Sankar and Vishwanathan
Combinatorial Nullstellensatz, by Alon

Beyond this, many excellent surveys on the polynomial method are available in which one can find further developments. For example,

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