The Oddtown problem modulo a composite number

TL;DR

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Contribution

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Abstract

A family of subsets $\mathcal{A}$ of an $n$-element set is called an $\ell$-Oddtown if the sizes of all sets are not divisible by $\ell$, but the sizes of pairwise intersections are divisible by $\ell$. Berlekamp and Graver showed that when is a $\ell$ is a prime, the maximum size of an $\ell$-Oddtown is $n$. For composite moduli with $\omega$ distinct prime factors, the argument of Szegedy gives an upper bound of $\omega n-\omega\log_2 n$ on the size of an $\ell$-Oddtown. We improve this to $\omega n-(2\omega +\varepsilon)\log_2 n$ for most $\ell$ and $n$ using a combination of linear algebraic and Fourier-analytic arguments.

Equations

$$f_{\ell }(n)\le \omega n.$$

$$f_{\ell }(n)\le \omega n-\omega \log n\text{for square-free }\ell ,$$

$$f_{\ell }(n)\le \omega n-2\omega \log n+11.$$

$$f_{\ell }(n)\le \omega n-(2\omega +\epsilon )\log n,$$

$$d\gtrsim \min \left(\left(1+\frac{1}{36p^2}\right)\log r,3\log c\right).$$

$$u_i\cdot u_i$$

$$u_i\cdot u_j$$

$$v_i\cdot v_j$$

$$u_i\cdot v_j$$

$$\dim ({\mathrm{span}}_{p_i}{\mathcal{A}}_i^{\prime })\le (n-\left|{\mathcal{A}}_i\right|)/2.$$

$$\left|{\mathcal{A}}_i^{\prime }\right|\le 2^{(n-|{\mathcal{A}}_i|)/2}.$$

$$u_i^{\prime }\cdot u_i^{\prime }$$

$$u_i^{\prime }\cdot u_j^{\prime }$$

$$u_i^{\prime }\cdot v_j^{\prime }$$

$$v_i^{\prime }\cdot v_i^{\prime }$$

$$v_i^{\prime }\cdot v_j^{\prime }$$

$$a_1{u}_1^{\prime }+\cdots +a_{n-k}{u}_{n-k}^{\prime }=b_1{v}_1^{\prime }+\cdots +b_m{v}_m^{\prime },$$

$$a_1{u}_1^{\prime }+\cdots +a_{n-k}{u}_{n-k}^{\prime }\equiv 0(\mathrm{mod}{p}^{\alpha }),$$

$${\mathcal{A}}_i$$

$${\mathcal{A}}_i^{\prime }:=\mathcal{A}\setminus {\mathcal{A}}_i$$

$$\left|\mathcal{A}\right|=\left|{\mathcal{A}}_i\right|+\left|{\mathcal{A}}_i^{\prime }\right|\le n+n/2\le \omega n-(2\omega +\epsilon )\log n$$

$$d\ge \left(1+\frac{1}{36p^2}\right)(\log k-2).$$

$$d_i\ge \left(1+\frac{1}{36p^2}\right)(\log \left|{\mathcal{A}}_i^{\prime }\right|-2)\ge \left(1+\frac{1}{36p_i^2}\right)(\log n-3)\ge \left(1+\frac{1}{36p_i^2}\right)\log n-4.$$

$$\left|{\mathcal{A}}_i\right|$$

$$\left|\mathcal{A}\right|\le \sum _{i=1}^{\omega }\left|{\mathcal{A}}_i\right|\le \omega n-\left(2\omega +\sum _{\text{good }{p}_i}\frac{1}{18p_i^2}\right)\log n+8\omega .$$

$$\left|\mathcal{A}\right|\le \left|{\mathcal{A}}_{j_i}\cap {\mathcal{A}}_{j_2}\right|+\left|{\mathcal{A}}_{j_1}^{\prime }\cup {\mathcal{A}}_{j_2}^{\prime }\right|\le n^{0.2}+\omega (n-n^{0.2}+\ell )\le \omega n-n^{0.2}+\omega \ell .\text{qed}$$

$$\sum _{\text{good }{p}_i}\frac{1}{p_i^2}\ge \sum _{p\in P_{\text{odd}}^{\ast }(\ell )}\frac{1}{p^2},$$

$$\left|\mathcal{A}\right|\le \omega n-(2\omega +\epsilon )\log n,$$

$$\mathbb{P}[L^{\top }v\in \{0,1{\}}^c]\le 3\cdot 2^{-d/36p^2}.$$

$$k\le 2^d\cdot 3\cdot 2^{-d/36p^2},$$

$$d\ge \left(1+\frac{1}{36p^2}\right)(\log k-2).\text{qed}$$

$$\hat{f}:{\mathbb{F}}_p^c\to \mathbb{C},\xi \mapsto \sum _{x\in {\mathbb{F}}_p^n}f(x)e^{-2\pi i⟨x,\xi ⟩/p},$$

$$\widehat{f_j}(\xi )=\sum _{x\in {\mathbb{F}}_p^c}{f}_j(x)e^{-2\pi i⟨x,\xi ⟩/p}=\frac{1}{2}\left(1+e^{-2\pi i⟨r_j,\xi ⟩/p}\right).$$

$$f(x)=\sum _{x_1+x_2+\cdots +x_d=x}{f}_1(x_1)f_2(x_2)\cdots f_d(x_d)=f_1\ast f_2\cdots \ast f_d(x),$$

Full text

Supported in part by U.S. taxpayers through NSF grants DMS-2154063, DMS-2452120 and a Simons Foundation Fellowship.

1. Introduction

The Oddtown problem is perhaps the simplest application of the linear algebra method in combinatorics, and it is often the first use of the method that students encounter. The problem asks for the largest size of a family $\mathcal{A}$ of subsets of $[n]:=\{1,2,\dotsc,n\}$ of odd cardinality such that intersection of any two set in the family is of even size. Original motivation for the question was a problem of Erdős that was solved independently by Berlekamp [4] and Graver [8]. The name of the problem comes from the book of Babai and Frankl [2], where it is cast as a story about a city whose council tries to contain prolifiration of clubs formed by its eccentic residents.

After identifying each set with its characteristic vector, simple mod-$2$ linear algebra shows that the family $\mathcal{A}$ can have no more than $n$ sets. The bound is sharp, as witnessed by the family of all single-element sets. There are many other sharp examples [2, Ex 1.1.14].

There have been a number of works examining applications of linear algebra modulo a composite number to combinatorics. A particularly notable one is the line of works by Grolmusz [10, 9] who constructed large families of sets with nice intersection properties modulo a composite number; these found numerous applications. Another adjacent recent line of works [1, 6, 5] concerns Kakeya problem over $\mathbb{Z}/{N\mathbb{Z}}$, which is a toy model for the original Kakeya problem.

Motivated by the desire to better understand linear algebra arguments in combinatorics, in this paper we examine the generalization of the Oddtown problem to composite moduli. By $\ell$-Oddtown we mean a set family $\mathcal{A}\subset 2^{[n]}$ satisfying

• •

$\left|A\right|\not\equiv 0\pmod{\ell}$ for any $A\in\mathcal{A}$, and

• •

$\left|A\cap B\right|\equiv 0\pmod{\ell}$ for any distinct $A,B\in\mathcal{A}$.

Let $f_{\ell}(n)$ denote the size of the largest $\ell$-Oddtown.

Known bounds.

Let $\omega$ be the number of distinct prime factors of $\ell$. If $\omega=1$, i.e., $\ell$ is a power of a prime, the same sharp bound $f_{\ell}(n)\leq n$ can be proved using basic linear algebra [3].

More generally, let $\ell=p_{1}^{\alpha_{1}}\dotsb p_{\omega}^{\alpha_{\omega}}$ be the prime factorization of $\ell$, and set $\mathcal{A}_{i}:=\{A\in\mathcal{A}:\left|A\right|\not\equiv 0\pmod{p_{i}^{e_{i}}}\}$; then $\left|\mathcal{A}_{i}\right|\leq n$ by the prime-power modulus bound just mentioned, which in view of $\mathcal{A}=\mathcal{A}_{1}\cup\dotsb\cup\mathcal{A}_{\omega}$ implies what we call the trivial bound

[TABLE]

As already mentioned, for $\omega=1$ this bound is sharp. So, for the rest of the paper we assume that $\omega\geq 2$.

The only improvement to the trivial bound that we are aware of is due to Mario Szegedy [2, Ex 1.1.28]. The statement appearing in [2] is for $\ell=6$ only, but the suggested argument generalizes to

[TABLE]

where, both here and hereafter, $\log$ denotes the logarithm to base $2$.

New result.

We more than double the $\omega\log n$ saving, and extend the result to general (non-square-free) moduli.

Theorem 1.1**.**

Let $\ell$ be a positive integer with at least two distinct prime factors. Then

[TABLE]

Theorem 1.2**.**

Let $\ell$ be a positive integer with at least two distinct odd prime factors. If $n$ is sufficiently large compared to $\ell$, then

[TABLE]

where $\varepsilon=\frac{1}{20}(\sum_{p\in P_{\text{odd}}^{*}(\ell)}p^{-2})$, and $P_{\text{odd}}^{*}(\ell)$ is the set of all odd primes $p$ dividing $\ell$ except for the smallest such prime.

The result is a combination of two new ingredients. First, we observe that in Szegedy’s argument a certain pairs of subspaces is orthogonal to each other. This permits us to replace $\omega\log n$ by $2\omega\log n$, but only for square-free moduli. The main difficulty is to generalize the orthogonality argument from a prime to prime-power moduli. This is done in Lemma 2.1.

Ranks of [math]–$1$ matrices with distinct rows and distinct columns.

The bulk of the paper is spent improving this bound by further $\varepsilon\log n$. In the proof of 1, a key role is played by the bound $\left|\{0,1\}^{n}\cap V\right|\leq 2^{d}$ that holds for any vector subspace $V$ of $\mathbb{F}_{p}^{n}$ of dimension $d$. Equivalently, if $M\in\mathbb{F}_{p}^{r\times c}$ is a [math]–$1$ matrix with distinct rows, then $d:=\operatorname{rank}_{p}M$ is always lower bounded by $\log r$.

It follows that, if both rows and columns of $M$ are distinct, then $d\geq\max(\log r,\log c)$. The main ingredient behind the $\varepsilon\log n$ improvement is the following result that excludes the possibility that both $\log r$ and $\log c$ are close to $d$ in this case.

Lemma 3.1** (Informal version).**

Let $p$ be an odd prime and let $M\in\mathbb{F}_{p}^{r\times c}$ be a [math]–$1$ matrix with distinct columns and distinct rows. Set $d:=\operatorname{rank}_{p}M$. Then we have

[TABLE]

The improvement is depicted in the following figure.

It is not difficult to construct $r$-by-$c$ matrices, for any $r$ and $c$, with distinct rows and distinct columns of rank at most $\approx\log r+\log c$. Indeed, let $M_{n}\in\mathbb{F}_{p}^{n\times 2^{n}}$ be the [math]–$1$ matrix whose columns are all possible [math]–$1$ vectors of length $n$. Then $M=\left(\begin{smallmatrix}M_{a}^{\top}&0\\ 0&M_{b}\end{smallmatrix}\right)$ is an $(2^{a}+b)$-by-$(a+2^{b})$ matrix. Since $\operatorname{rank}_{p}M_{n}=n$, the matrix $M$ is of rank $a+b$.

We suspect that the fraction $1/36p^{2}$ in Lemma 3.1 can be improved to some absolute constant $\varepsilon^{\prime}>0$. This would imply that the constant $\varepsilon>0$ in Theorem 1.2 can be made independent of $\ell$.

Paper organization.

The remainder of this paper is structured as follows. In Section 2 we introduce our algebraic tools, and prove Theorem 1.1. In Section 3 we present the proof of our main result Theorem 1.2, modulo the proof of Lemma 3.1, which is postponed to Section 4. Lemma 3.1 is a combination of the Fourier-analytic argument in Proposition 4.2 and a combinatorial covering argument in Lemma 4.3. Unlike the combinatorial part of the argument, the Fourier-analytic component is rather unsophisticated. However, it is interesting because the applications of Fourier analysis to extremal set theory are very rare. The most recent example is probably an approach to the density Hales–Jewett theorem by Gowers and Karam [7].

2. Algebraic part: proof of the $2\omega\log n$ bound

The following lemma is easy for prime moduli using the usual linear algebra in the prime field $\mathbb{F}_{p}$. The fact that it holds for any prime power is the key to proving Theorem 1.1 in full generality.

Lemma 2.1**.**

Suppose that $u_{1},\ldots,u_{n-k}\in\mathbb{Z}^{n}$ and $v_{1},\dotsc,v_{m}\in\mathbb{Z}^{n}$ are vectors satisfying

[TABLE]

Then the vector space spanned by $v_{1}\bmod{p},\ldots,v_{m}\bmod{p}$ over $\mathbb{F}_{p}$ has dimension at most $k/2$.

We postpone the proof of this lemma until the end of this section. For now, we show how to use the lemma to prove that every $\ell$-Oddtown $\mathcal{A}\subset 2^{[n]}$ satisfies $\left|\mathcal{A}\right|\leq\omega n-2\omega\log n+11$.

We need the following well-known fact (appearing, for example, in [11]).

Lemma 2.2**.**

Let $\mathbb{F}$ be a field, and $V\subset\mathbb{F}^{n}$ be a subspace of dimension $d$. Then $\left|V\cap\{0,1\}^{n}\right|\leq 2^{d}$.

Proof.

Let $M$ be a $d$-by-$n$ matrix of rank $d$ whose rows span $V$. Since the row rank is equal to the column rank, some set of $d$ columns, say the first $d$ columns, span the column space of $M$. This means that every point in $V$ is determined by its first $d$ coordinates, which implies the lemma. ∎

Let $\ell=p_{1}^{\alpha_{1}}\dotsb p_{\omega}^{\alpha_{\omega}}$ be the prime factorization of $\ell$. Recall that $\mathcal{A}_{i}:=\{A\in\mathcal{A}:\left|A\right|\nequiv 0\pmod{p_{i}^{\alpha_{i}}}\}$, and set $\mathcal{A}_{i}^{\prime}:=\mathcal{A}\setminus\mathcal{A}_{i}$.

Fix any $i\in[\omega]$, and apply Lemma 2.1 with the characteristic functions of sets in $\mathcal{A}_{i}$ and in $\mathcal{A}_{i}^{\prime}$ playing the role of vectors $u_{1},\dotsc,u_{n-k}$ and $v_{1},\dotsc,v_{m}$ respectively. We learn that

[TABLE]

In view of Lemma 2.2, this implies that

[TABLE]

If $\omega\geq 3$ and $\left|\mathcal{A}_{i}^{\prime}\right|\leq n$ for some $i$, then $\left|\mathcal{A}\right|\leq\left|\mathcal{A}_{i}\right|+\left|\mathcal{A}_{i}^{\prime}\right|\leq 2n\leq\omega n-2\omega\log n+11$. If $\omega\geq 3$ and $\left|\mathcal{A}_{i}^{\prime}\right|\geq n$ for all $i$, then (4) implies that $\left|\mathcal{A}\right|\leq\sum_{i}\left|\mathcal{A}_{i}\right|\leq\sum_{i}(n-2\log n)=\omega n-2\omega\log n$.

Similarly, if $\omega=2$ and $\left|\mathcal{A}_{i}^{\prime}\right|\leq n/2$ for some $i=1,2$, then $\left|\mathcal{A}\right|\leq\tfrac{3}{2}n\leq 2n-4\log n+11$. Otherwise, for both $i=1,2$ we have $\left|\mathcal{A}_{i}\right|\leq n-2\log\left|\mathcal{A}^{\prime}_{i}\right|\leq n-2\log n+2$, implying $\left|\mathcal{A}\right|\leq 2n-4\log n+4$ in this case. So, in the either case $\left|\mathcal{A}\right|\leq 2n-4\log n+11$ holds.

Proof of Lemma 2.1..

We first observe that the properties of the $u_{i}$ and $v_{j}$ vectors are preserved under the following operations:

1. (i)

Replacing any vector by its sum or difference with a $v_{j}$; 2. (ii)

Multiplying any vector by an integer coprime to $p$; 3. (iii)

Adding a multiple $p^{\alpha}w$ ($w\in\mathbb{Z}^{n}$) to any vector.

Let $V:=\operatorname{span}_{p}\{v_{1}\bmod p,\dotsc,v_{m}\bmod p\}$. By removing some of the vectors, we may assume without loss of generality that $v_{1}\bmod p,\dotsc,v_{m}\bmod p$ form a basis for $V$, so that $m=\dim_{p}V$. By Gaussian elimination applied to these $m$ vectors, there exists $m$ coordinates on which $V$ projects bijectively. We may assume that these are the first $m$ coordinates. Using operation (i), we can also assume that for any $s,t\in[m]$,$v_{s}^{(t)}\not\equiv 0\pmod{p}$ if and only if $s=t$, where $v_{s}^{(t)}$ denotes the $t$-th coordinate of $v_{s}$. Moreover, note that for any integer $z\nequiv 0\pmod{p}$, there exists $z^{\dagger}$ such that $zz^{\dagger}\equiv 1\pmod{p^{\alpha}}$. We can therefore use operations (ii) and (iii) to ensure that $v_{s}^{(t)}=\delta_{s,t}$.

We may then apply operation (i) to zero out $u_{i}^{(s)}$, for any $i\in[n-k]$ and any $s\in[m]$, by subtracting a suitable multiple of $v_{s}$. Hence, we may assume that $u_{i}^{(s)}=0$ for all $i\in[n-k]$ and all $s\in[m]$.

Let $u^{\prime}_{i}:=u_{i}|_{[n]\setminus[m]}$ and $v^{\prime}_{i}:=v_{i}|_{[n]\setminus[m]}$ denote the projections of the vectors $u_{i}$ and $v_{j}$ onto the remaining $n-m$ coordinates. The assumptions (2) imply that

[TABLE]

Supposing on the contrary that $n-m<n+m-k$, we deduce that $\{u_{i}^{\prime},v_{j}^{\prime}\}_{i\in[n-k],j\in[m]}$ is linearly dependent over $\mathbb{Z}$ because they are vectors in $\mathbb{Z}^{n-m}$. Let

[TABLE]

be a linear dependence, with $a_{i},b_{j}\in\mathbb{Z}$. We may assume that $p$ does not divide all $a_{i}$’s and all $b_{j}$’s. Taking dot product of both sides with $v_{j}^{\prime}$ yields $b_{j}\equiv 0\pmod{p^{\alpha}}$, for any $j\in[m]$. Hence,

[TABLE]

is a non-trivial linear combination of the vectors $u_{1}^{\prime},\dotsc,u_{n-k}^{\prime}$ modulo $p^{\alpha}$. Upon taking the dot product with $u_{j}^{\prime}$, for any $j$ satisfying $a_{j}\nequiv 0\pmod{p}$, we reach a contradiction. ∎

3. Proof of Theorem 1.2

Let $\mathcal{A}$ be an $\ell$-Oddtown on ground set $[n]$. Recall the definitions, for $i\in[\omega]$,

[TABLE]

Similar to the proof of Theorem 1.1 in Section 2, if $\left|\mathcal{A}^{\prime}_{i}\right|\leq n/2$, then

[TABLE]

when $n$ is large enough, and we are done in this case. Hence, we may assume that $\left|\mathcal{A}^{\prime}_{i}\right|>n/2$ from now on.

Let $M_{i}$ be the $\left|\mathcal{A}_{i}^{\prime}\right|$-by-$n$ matrix whose rows are the characteristic vectors of sets in $\mathcal{A}_{i}^{\prime}$. The following lemma (whose proof we defer) is, if its conditions are satisfied, a strenghtening of Lemma 2.2 that we used in the proof of Theorem 1.1. We wish to apply this lemma to the matrices $M_{i}$, when $p_{i}$ is odd.

Lemma 3.1** (Main lemma).**

Let $p$ be an odd prime and let $M\in\mathbb{F}_{p}^{k\times c}$ be a [math]–$1$ matrix with distinct columns and distinct rows. Set $d=\operatorname{rank}_{p}M$ and assume that $36\log p\leq d\leq 3\log c$. Then we have

[TABLE]

However, although the rows of $M_{i}$ are distinct, the columns might be repeated. Therefore, we shall apply Lemma 3.1 to the submatrix $M_{i}^{\prime}$ of $M_{i}$ obtained by deleting repeated columns. We say that the odd prime $p_{i}$ dividing $\ell$ is good if $M_{i}$ has at least $n^{0.4}$ distinct columns, otherwise we say that $p_{i}$ is bad.

If $p_{i}$ is good, then $d_{i}:=\operatorname{rank}_{p_{i}}(M_{i})$ satisfies one of the three possible inequalities:

• •

either $d_{i}\leq 36\log p_{i}$,

• •

or $d_{i}\geq(1+1/36p_{i}^{2})(\log\left|\mathcal{A}^{\prime}_{i}\right|-2),$

• •

or $d_{i}\geq 3\log(n^{0.4})$.

In the first case, $\left|\mathcal{A}^{\prime}_{i}\right|\leq 2^{d_{i}}\leq p_{i}^{36}$ contradicting our assumptions that $\left|\mathcal{A}^{\prime}_{i}\right|>n/2$ and that $n$ is large enough. In the second case,

[TABLE]

In the third case, $d_{i}\geq 1.2\log n\geq(1+1/36p_{i}^{2})\log n$, and so (5) holds in this case, too.

Combining (5) with the inequality $n-\left|\mathcal{A}_{i}\right|\geq 2d_{i}$ from (3) in Section 2 leads to

[TABLE]

Summing the inequalities above over all prime divisors of $\ell$, we get

[TABLE]

The following lemma shows that most odd prime divisors of $\ell$ are good, otherwise we get a better upper bound.

Lemma 3.2**.**

Either at most one odd prime divisor of $\ell$ is bad, or $\left|\mathcal{A}\right|\leq\omega n-n^{0.2}+\omega\ell$.

Proof.

Suppose that odd primes $p_{j_{1}},p_{j_{2}}$ are both bad. If $\left|\mathcal{A}_{j_{1}}\cap\mathcal{A}_{j_{2}}\right|\geq n^{0.2}$, then we use the trivial bound $\left|\mathcal{A}_{i}\right|\leq n$ to deduce that $\left|\mathcal{A}\right|=\left|\bigcup_{i=1}^{\omega}\mathcal{A}_{i}\right|\leq\sum_{i=1}^{\omega}\left|\mathcal{A}_{i}\right|-\left|\mathcal{A}_{j_{i}}\cap\mathcal{A}_{j_{2}}\right|\leq\omega n-n^{0.2}$.

Therefore, we may assume that $\left|\mathcal{A}_{j_{1}}\cap\mathcal{A}_{j_{2}}\right|\leq n^{0.2}$. Since $j_{1}$ is bad, there are at most $n^{0.4}$ distinct columns in the matrix $M_{j_{1}}$. Therefore, there is a set of elements $S^{\prime}\subseteq[n]$ of size at least $n^{0.6}$ such that the columns of $M_{j_{1}}$ indexed by $S^{\prime}$ are all the same. Similarly, there is a subset $S\subseteq S^{\prime}$ of size $n^{0.2}$ such that the columns of $M_{j_{2}}$ indexed by $S$ are all the same.

Note that the set family $\mathcal{B}:=\mathcal{A}^{\prime}_{j_{1}}\cup\mathcal{A}^{\prime}_{j_{2}}$ is also an $\ell$-Oddtown. By construction, all elements of $S$ are contained in exactly the same collection of sets in $\mathcal{B}$. Therefore, we may delete any $\ell$ elements of $S$ from all the sets in $\mathcal{B}$ without changing the $\ell$-Oddtown property. By repeating this until fewer than $\ell$ elements in $S$ remain, we obtain an $\ell$-Oddtown supported on at most $n-\left|S\right|+\ell\leq n-n^{0.2}+\ell$ elements. By the trivial bound, we have $\left|\mathcal{A}^{\prime}_{j_{1}}\cup\mathcal{A}^{\prime}_{j_{2}}\right|\leq\omega(n-n^{0.2}+\ell)$. Thus,

[TABLE]

By this lemma, we may assume without loss of generality that at most one odd prime divisor of $\ell$ is bad. Therefore,

[TABLE]

and hence

[TABLE]

when $n$ is large enough.

4. Analytic part: Fourier-analytic argument

Reduction to a strengthing of Lemma 2.2.

In this section, we prove Lemma 3.1. We do this by reducing it to the following result.

Lemma 4.1** (Main lemma, probabilistic form).**

Let $p$ be an odd prime and let $c,d$ be positive integers such that $36\log p\leq d\leq 3\log c$. Let $L\in\mathbb{F}_{p}^{d\times c}$ be a matrix whose columns are distinct vectors in $\mathbb{F}_{p}^{d}$, and let $v$ be uniformly distributed over $\{0,1\}^{d}$. Then

[TABLE]

Proof of Lemma 3.1 assuming Lemma 4.1.

Let $r_{1},\dots,r_{k}$ be the row vectors of $M$. Since the rank of $M$ is $d$, we may assume without loss of generality that $r_{1},\dots,r_{d}$ span the row space. Furthermore, we may pick $d$ columns such that the vectors $r_{1},\dots,r_{d}$ restricted to these columns are linearly independent. Assume without loss of generality that these columns are the first $d$ columns. Let $r^{\prime}_{1},\dots,r^{\prime}_{d}$ be the linear combinations of $r_{1},\dots,r_{d}$ such that the projections of $r^{\prime}_{1},\dots,r^{\prime}_{d}$ to the first $d$ columns form the identity matrix.

Now, let $L\in\mathbb{F}_{p}^{d\times c}$ denote the matrix with rows $r^{\prime}_{1},\dots,r^{\prime}_{d}$. Consider any row $r_{x}$ of $L$; since $r_{x}$ is a linear combination of $r_{1},\dots,r_{d}$, it is also a linear combination of $r^{\prime}_{1},\dots,r^{\prime}_{d}$ as well, say $r_{x}=\sum_{i=1}^{d}a_{i}r^{\prime}_{i}$. Note that the coefficients $a_{1},\dotsc,a_{d}$ are in $\{0,1\}$ because the $i$-th coordinate of $r_{x}$ is $a_{i}$, for all $i\in[d]$. Therefore, the vector $r_{x}$ is of the form $v^{\top}L$ with $v\in\{0,1\}^{d}$.

Note that since $L$ can be obtained from $M$ by a sequence of elementary row operations, and the columns of $M$ are distinct, it follows that the columns of $L$ too are distinct.

Lemma 4.1 says that the number of rows of $M$ satisfies

[TABLE]

and hence

[TABLE]

Proof of the main lemma.

We begin by obtaining bounds on the Fourier transform of the probability distribution induced by the action of distinct linear forms on random [math]–$1$ vectors. These Fourier-analytic estimates will enable us to control the number of [math]–$1$ vectors that lie in subspaces of $\mathbb{F}_{p}^{c}$.

Let $p$ be a prime number. For any function $f\colon\mathbb{F}_{p}^{c}\to\mathbb{C}$, its Fourier transform is defined by

[TABLE]

where the inner product $\langle x,\xi\rangle$ takes values in $(-p/2,p/2]$.

Proposition 4.2**.**

Let $p$ be a prime number and let $L\in\mathbb{F}_{p}^{d\times c}$ be a matrix with distinct columns. Suppose that $v\in\{0,1\}^{d}$ is chosen uniformly at random and let $f(x)$ denote the probability that $L^{\top}v=x$. Then, for every $\xi\in\mathbb{F}_{p}^{c}$, we have $\lvert\hat{f}(\xi)\rvert\leq e^{-\frac{\pi^{2}\sigma(\xi)}{2p^{2}}}$, where $\sigma(\xi):=\sum_{j=1}^{d}\langle r_{j},\xi\rangle^{2}$ and $r_{j}$ denotes the $j$-th row of $L$.

Proof.

Let $f_{j}(x)$ denote the probability that $r_{j}^{\top}v^{(j)}=x$. Since $f_{j}(0)=f_{j}(r_{j}^{\top})=1/2$ and $f_{j}(x)=0$ for $x\notin\{0,r_{j}^{\top}\}$, the Fourier transform of $f_{j}$ is

[TABLE]

Note that

[TABLE]

where $*$ denotes convolution. Consequently, the Fourier transform of $f(x)$ satisfies

[TABLE]

Taking absolute values and using the inequality $|(1+e^{-2\theta i})/{2}|=|\cos(\theta)|\leq e^{-\theta^{2}/2}$ that is valid for $\theta\in(-\pi/2,\pi/2]$, we obtain

[TABLE]

We say a matrix $S\in\mathbb{F}_{p}^{d\times c}$ with rows $r_{1},r_{2},\ldots,r_{d}$ is $\sigma$-admissible if for any nonzero $\xi\in\mathbb{F}^{c}$,

[TABLE]

We also write $H(q):=-q\log q-(1-q)\log(1-q)$ for the binary entropy function.

Note that if $S$ is $\sigma$-admissible, then any submatrix $S^{\prime}\in\mathbb{F}_{p}^{d\times c^{\prime}}$ is also $\sigma$-admissible since we can set the entries of $\xi$ corresponding to the columns that are not in $S^{\prime}$ to zero.

Lemma 4.3**.**

Let $p$ be an odd prime number, $\sigma>0$ be a real number, and $L\in\mathbb{F}_{p}^{d\times c}$ be a matrix with distinct column vectors. There exists a $\sigma$-admissible submatrix $S\in\mathbb{F}_{p}^{d\times c^{\prime}}$ of $L$ with

[TABLE]

Proof.

Consider all submatrices of $L$ that are obtained from $L$ by deleting some columns. Let $S\in\mathbb{F}_{p}^{d\times c^{\prime}}$ be a maximal $\sigma$-admissible such submatrix. If there is no such submatrix, we set $c^{\prime}=0$ and $S$ to be the empty matrix.

Suppose $w$ is a column of $L$ that is not in $S$, and let $S^{\prime}$ be the matrix obtained by appending $w$ to $S$. It is not $\sigma$-admissible by the maximality of $S$. This implies the existence of a vector $\eta\in\mathbb{F}_{p}^{c^{\prime}+1}$ such that

[TABLE]

where $r^{\prime}_{1},\dots,r^{\prime}_{d}$ are the row vectors of $S^{\prime}$.

We claim that $\eta_{c^{\prime}+1}\neq 0$. Indeed, if $\eta_{c^{\prime}+1}=0$ were true, then the restriction of $\eta$ to the first $c^{\prime}$ coordinates would satisfy $\sigma\geq\sum_{i=1}^{d}\langle r_{i}^{\prime},\eta\rangle^{2}=\sum_{i=1}^{d}\langle r_{i},\operatorname{proj}(\eta)\rangle^{2}$, where $\operatorname{proj}(\eta)$ denotes the projection onto the first $c^{\prime}$ coordinates; this would contradict the $\sigma$-admissibility of $S$. This observation bounds the number of possible choices for $\eta$ by $(p-1)p^{c^{\prime}}$. Note that this also holds in the case where $c^{\prime}=0$ since $\eta$ must be a non-zero vector.

Next, we upper bound the number of all possible $w$. For fixed $\eta$ satisfying $\eta_{c^{\prime}+1}\neq 0$, the function $w\mapsto(\langle r_{1}^{\prime},\eta\rangle,\dotsc,\langle r_{d}^{\prime},\eta\rangle)$ is injective. That is because the $i$-th coordinate of $w$ is determined by $\langle r_{i}^{\prime},\eta\rangle$ and $r_{i}$, the latter of which is fixed because $S$ is fixed. By (6), the vector $(\langle r_{1}^{\prime},\eta\rangle,\dotsc,\langle r_{d}^{\prime},\eta\rangle)$ satisfies the inequality

[TABLE]

Since the number of solutions to the inequality $\sum_{i=1}^{d}y_{i}\leq\sigma$ in nonnegative integers $y_{1},\dotsc,y_{d}$ is $\binom{\lfloor\sigma\rfloor+d}{d}\leq\nobreak\binom{\sigma+d}{\sigma}$, it follows that the number of vectors $(x_{1},x_{2},\ldots,x_{d})$ satisfying (7) is at most $2^{\sigma}\binom{\sigma+d}{\sigma}$. Therefore, the number of column vectors $w$ in $L$ that are not in $S$, which equals $c-c^{\prime}$, is at most $(p-1)p^{c^{\prime}}2^{\sigma}\binom{\sigma+d}{\sigma}$.

Therefore,

[TABLE]

and hence

[TABLE]

With these results in hand, we are now ready to complete the proof of our main lemma.

Proof of Lemma 4.1.

Upon applying Lemma 4.3 with $\sigma=d/36$, we learn that there exists a $\sigma$-admissible submatrix $S\in\mathbb{F}_{p}^{d\times c^{\prime}}$ of $L$, where

[TABLE]

Here, we used the fact that $\frac{1}{36}+\frac{37}{36}H\left(\frac{1}{37}\right)\leq\frac{1}{4}$.

Let $T\in\mathbb{F}_{p}^{d\times c^{\prime\prime}}$ be any submatrix of $S$ with $c^{\prime\prime}$ columns, where

[TABLE]

Note that $T$ is also $\sigma$-admissible.

Let $f(x)$ denote the probability mass function of $T^{\top}v$. By the Fourier inversion formula and the bound on the Fourier transforms from Proposition 4.2,

[TABLE]

Therefore,

[TABLE]

We may bound these two terms by

[TABLE]

and

[TABLE]

Here, we used the estimates $(\log p-1)/\log p>1/3$ and $e^{\pi^{2}/36}\geq 2^{1/6}$. Thus, we may conclude that

[TABLE]