On Szemer\'edi's theorem with differences from a random set

TL;DR

This paper investigates Szemerédi's theorem with restricted, random differences over integers and finite fields, linking thresholds to conjectures about dual functions and nilsequences, revealing different behaviors in these settings.

Contribution

It connects the threshold for random differences in Szemerédi's theorem to dual function approximations by nilsequences and compares integer and finite field cases.

Findings

Threshold over integers depends on dual function approximation conjecture.

Threshold over finite fields differs from that over integers.

Provides new insights into the behavior of arithmetic progressions with random differences.

Abstract

We consider, over both the integers and finite fields, Szemer\'{e}di's theorem on $k$-term arithmetic progressions where the set $S$ of allowed common differences in those progressions is restricted and random. Fleshing out a line of enquiry suggested by Frantzikinakis et al, we show that over the integers, the conjectured threshold for $\mathbb{P}(d \in S)$ for Szemer\'{e}di's theorem to hold a.a.s follows from a conjecture about how so-called dual functions are approximated by nilsequences. We also show that the threshold over finite fields is different to this threshold over the integers.

Equations

$$F_A(d):={\mathbb{E}}_x{1}_A(x)1_A(x+d)1_A(x+2d),$$

$$\mathbb{P}(d\in S)=\frac{c{n}^2}{p^n},$$

$$F_A^{(2)}(d):={\mathbb{E}}_x{1}_A(x)1_A(x+d),$$

$$⟨F_A^{(2)},1_S⟩:=$$

$$=$$

$$F_A(d):={\mathbb{E}}_x{1}_A(x)1_A(x+d)1_A(x+2d),$$

$$F_A(d)=\sum _j{c}_{A,j}{\varphi }_j(d)+e_A(d).$$

$$\mathbb{P}\left(\underset{A}{\sup }\left|⟨F_A,Y⟩\right|⩾6\epsilon \sigma \right)=o(N^{-100}).$$

$${\mathbb{E}}_{x,d}{1}_A(x)1_A(x+d)1_A(x+2d)1_S(d)>\sigma {\mathbb{E}}_{x,d}{1}_A(x)1_A(x+d)1_A(x+2d)-6\epsilon \sigma$$

$${\mathbb{E}}_{x,d}{1}_A(x)1_A(x+d)1_A(x+2d)1_S(d)>0,$$

$$F_A=\sum _j{c}_{A,j}{\varphi }_j+e_A,$$

$$\mathbb{P}\left(\underset{A}{\sup }\left|⟨F_A,Y⟩\right|⩾6\epsilon \sigma \right)$$

$$⩽\mathbb{P}\left(\underset{j}{\sup }\left|⟨{\varphi }_j,Y⟩\right|⩾3\epsilon \sigma \right)$$

$$+\mathbb{P}\left(\underset{A}{\sup }\left|⟨e_A,Y⟩\right|⩾3\epsilon \sigma \right).$$

$$\mathbb{P}\left(\underset{j}{\sup }\left|⟨{\varphi }_j,Y⟩\right|⩾3\epsilon \sigma \right)⩽N^{O_{\epsilon }(1)}\mathbb{P}\left(\left|⟨\varphi ,Y⟩\right|⩾3\epsilon \sigma \right),$$

$$\mathbb{P}\left(\left|⟨\varphi ,Y⟩\right|⩾3\epsilon \sigma \right)⩽\exp \left(-C_{\epsilon }\sigma N\right).$$

$$\mathbb{P}\left(\underset{j}{\sup }\left|⟨{\varphi }_j,Y⟩\right|⩾3\epsilon \sigma \right)$$

$$=o(N^{-100}),$$

$$\mathbb{P}\left(\underset{A}{\sup }\left|⟨e_A,Y⟩\right|⩾3\epsilon \sigma \right)$$

$$⩽\mathbb{P}\left(\sum _{d=1}^N\left(\left|Y(d)\right|-2\sigma (1-\sigma )\right)⩾N\sigma (1+2\sigma )\right)$$

$$⩽\exp \left(-CN\sigma \right)$$

$$=o(N^{-100}).$$

$$\mathbb{P}\left(\underset{A}{\sup }|⟨F_A,Y⟩|⩾6\epsilon \sigma \right)=o(N^{-100}),$$

$$\mathbb{P}\left(\underset{A}{\sup }\left|⟨F_A,Y⟩\right|⩾6\epsilon \sigma \right)=o(N^{-100}),$$

$$\left|{\mathbb{E}}_d\mathbb{E}(\varphi (d)Y(d))\right|=o(\sigma ),$$

$$\mathbb{P}(G_m^c)=\mathbb{P}\left(\bigcup _{B\in {\mathrm{limsup}}_N{I}_{m,N}}{E}_B^c\right)⩽\mathbb{P}\left(\underset{N}{\mathrm{limsup}}\bigcup _{B\in I_{m,N}}{E}_{B,N}^c\right).$$

$$\mathbb{P}(d\in S)=\frac{c{n}^2}{p^n}=\frac{c{\log }_p^2|{\mathbb{F}}_p^n|}{|{\mathbb{F}}_p^n|},$$

$$\left|\bigcup _{i=1}^k{v}_i^{\perp }\right|=p^m\left(1-{\left(\frac{p-1}{p}\right)}^k\right).$$

$$\bigcup _{i=1}^k{v}_i^{\perp }⊊{\mathbb{F}}_p^m.$$

$${\left(1-{\mathbb{P}}_{d\in {\mathbb{F}}_p^n}\left(\phi (d)\in W_N\right)\right)}^N.$$

$$\mathbb{P}(d_1\ne 0)\prod _{i=2}^N\mathbb{P}(\phi (d_i)\notin \mathrm{span}\{\phi (d_1),\dots ,\phi (d_{i-1})\})$$

$$⩾$$

$${\mathbb{P}}_{d\in {\mathbb{F}}_p^n}\left(\phi (d)\in W_N\right)⩽{\mathbb{E}}_{v\in W_N^{\perp }}{p}^{-\frac{1}{2}\mathrm{rank}{M}_v}.$$

$${\mathbb{P}}_x(V(x)=0)={\mathbb{E}}_{x,v}{\omega }^{v(x)}.$$

Full text

1. Introduction

1.1. Notation and definitions

For a positive integer $N$, let $[N]$ denote the set $\{1,\ldots,N\}$.

For $\delta>0$, a subset of $A$ of $[N]$ (respectively $\mathbb{F}_{p}^{n}$) will be said to have $\delta$-positive density (or be $\delta$-dense) if $|A|\geqslant\delta N$ (respectively $\geqslant\delta p^{n}$). A subset $B$ of $\mathbb{N}$ will be said to have positive upper density if $\limsup_{N}|B\cap[N]|/N>0$.

In a subset $A$ of an abelian group , a $k$-term arithmetic progression in $A$ (also $k$AP) is a pair $(x,d)$ such that $x,x+d,\ldots,x+(k-1)d\in A$. For $S\subset\mathbb{N}$, a $k$-term arithmetic progression has common difference in $S$ if, in the above notation, $d\in S$.

When the ambient set is $[N]$ (respectively $\mathbb{F}_{p}^{n}$), we will say that Szemerédi’s theorem with common differences in $S$ holds if, for all $\delta>0,k\geqslant 2,N>N_{0}(k,\delta)$ (respectively $n>n_{0}(k,\delta)$) and sets $A\subset[N]$ (respectively $\subset\mathbb{F}_{p}^{n}$) of $\delta$-positive density, there exists a non-trivial $k$-term arithmetic progression in $A$ with common difference in $S$. When the ambient set is $\mathbb{N}$, Szemerédi’s theorem with common differences in $S$ holds means that all sets $B\subset\mathbb{N}$ with positive upper density contain a non-trivial $k$-term arithmetic progression with common difference in $S$.

For a finite set $T$, we use the notation $\mathbb{E}_{x\in T}$ to denote the average over $T$, that is, $\frac{1}{|T|}\sum_{x\in T}$. We will often suppress the set $T$ and write $\mathbb{E}_{x}$ when the ambient set for $x$ is clear from context.

1.2. Context

In 1953, Roth [Rot53] showed that sets of integers with positive upper density contain non-trivial 3-term arithmetic progressions. The result was famously extended to arbitrarily long arithmetic progressions by Szemerédi in 1975 [Sze75]. It is well known that this is equivalent to the finitary formulation which asserts that, for $N$ sufficiently large in terms of $k$ and $\delta$, all $\delta$-dense subsets of $[N]$ contain non-trivial $k$APs.

A natural generalization is to consider under what conditions Szemerédi’s theorem is true when the set $S$ of allowed common differences in arithmetic progressions is restricted. It transpires that Szemerédi’s theorem holds with common differences restricted to some fairly sparse sets $S\subset\mathbb{N}$; for example, a result of Bergelson and Leibman [BL96] says that $S=\{1^{100},2^{100},3^{100},\ldots\}$ (or indeed $\{p(n):n\in\mathbb{N}\}$ for an integer polynomial $p$ with $p(0)=0$) is sufficient. Of course, the set $S$ under consideration by Bergelson and Leibman is of a special structure.

We are interested in the situation where $S$ is chosen at random. In the finitary model, it is common practice to construct the random set $S$ by selecting each $d$ to lie in $S$ independently with equal probability. In $\mathbb{N}$, the probability that $d$ lies in $S$ must be a function of $d$.

For $2$APs, it is known [Bou87] that if $\mathbb{P}(d\in S)=\omega(\log N/N)$ then Szemerédi’s theorem with common difference in $S$ holds asymptotically almost surely (a.a.s), and conversely that if $\mathbb{P}(d\in S)\leqslant C\log N/N$ then Szemerédi’s theorem with common difference in $S$ a.a.s fails. For $k$APs, the current best result is due to Briët and Gopi in [BG18], which states that $\mathbb{P}(d\in S)=\omega\left(\frac{\log N}{N^{1/\left\lceil k/2\right\rceil}}\right)$ is sufficient. In the case that $k=3$, this bound does not improve upon earlier work of Christ and Frantzikinakis, Lesigne and Wierdl in [Chr11] and [FLW12] respectively.

For $k\geq 3$, there is a substantial gap between these results and conjectures found in work of Frantzikinakis and others. We include a reformulation here for convenience.

Conjecture 1.1** ([Fra16, Problem 31], [FLW16, Conjecture 2.5]).**

Let $S\subset\mathbb{N}$ be chosen at random with $\mathbb{P}(d\in S)=\omega(1/d)$. Then it is almost surely the case that all subsets of $\mathbb{N}$ with positive upper density contain a $k$-term arithmetic progression with common difference in $S$.

Conjecture 1.1 is in fact best possible in the sense that if $S$ is constructed with $\mathbb{P}(d\in S)=1/d$, then Szemerédi’s theorem with common difference in $S$ fails (see discussion in [FLW16, Section 2]).

1.3. Our results

We study Szemerédi’s theorem with differences restricted to random subsets of $[N],\mathbb{N}$ and $\mathbb{F}_{p}^{n}$. Although many of our methods generalize straightforwardly, we will focus on $k=3$ as much still remains to be understood about this special case.

In Section 2 (over $[N]$), by analogy to the case $k=2$, Conjecture 2.1 stipulates that so-called dual functions

[TABLE]

for dense sets $A$ are well approximated by 2-step nilsequences. (This conjecture is similar to [Fra16, Special Case of Problem 1] - see [Fra16, Problem 1] for a discussion and related results). We show that under Conjecture 2.1, Szemerédi’s theorem in $[N]$ with common difference in $S$ a.a.s holds under two different probability models. Firstly, in Theorem 2.4, we choose $d$ to lie in $S$ with probability $\omega(\log N/N)$. Next, in Theorem 2.5, we choose $d$ to lie in $S$ with probability $\omega(1/d)$. The latter result is used to establish (almost surely) Szemerédi’s theorem in $\mathbb{N}$ with common difference in $S$ where $\mathbb{P}(d\in S)=\omega(1/d)$ (Corollary 2.6). Thus, our Conjecture 2.1 implies Conjecture 1.1 above (focusing on the case $k=3$).

In Section 3 we show (Corollary 3.2) that the analogous result to Theorem 2.4 over finite fields is false (by some margin). Indeed, if $S$ is formed by selecting elements with probability

[TABLE]

with $c=\frac{1}{2}-o(1)$, then Szemerédi’s theorem for $k=3$ with common difference in $S$ almost surely fails. We contrast this behavior with the case $k=2$, where the threshold for $\mathbb{P}(d\in S)$ for Szemerédi’s theorem in $[N]$ to hold is analogous to the threshold over finite fields.

2. Over the integers

The result that $\mathbb{P}(d\in S)=\omega(\log N/N)$ is sufficient for Szemerédi’s theorem on 2APs to a.a.s hold in $[N]$ (see e.g. [Bou87]) can be proven by considering the 2-dual functions

[TABLE]

which count the average number of $2$APs in dense sets $A$ with common difference $d$. Then one is interested in the quantity

[TABLE]

which counts the average number of 2APs in $A$ with common difference in $S$. The functions $F_{A}^{(2)}$ are just convolutions and so, by Fourier analysis, are well-approximated by convex combinations of additive characters selected uniformly in $A$. Then, roughly speaking, one can control the quantity $\langle F_{A}^{(2)},1_{S}\rangle$ uniformly in $A$ by controlling $\langle\phi,1_{S}\rangle$ across a set of additive characters $\{\phi\}$.

In pursuit of a similar argument for the case $k=3$, we define

[TABLE]

and provide Conjecture 2.1, that these dual functions are well-approximated by convex combinations of 2-step nilsequences.

See [Fra16, Problem 1] for similar conjectures and related discussion. See also Appendix A for details on what we mean by a nilsequence and its complexity.

Conjecture 2.1**.**

Let $\varepsilon>0$. There exists a set of 2-step nilsequences $\{\phi_{j}\}$ of complexity $O_{\varepsilon}(1)$, and, for every $A$, a family of coefficients $c_{A,j}$ with $\sum_{j}|c_{A,j}|\leqslant 1$ and a family of error functions $e_{A}$ with $\left|e_{A}(d)\right|\leqslant\varepsilon$ for $d=1,\ldots,N$, such that

[TABLE]

The difficulty in proving Conjecture 2.1 lies in controlling the error $e_{A}$ in $L^{\infty}$-norm. For example, if we only demanded control of $e_{A}$ in $L^{2}$-norm, then the above decomposition would follow as a consequence of the Koopman–von Neumann theorem.

The following proposition (known to experts) will yield that it suffices to consider only polynomially-many nilsequences. We defer to Appendix A for a proof.

Proposition 2.2**.**

Let $s,\varepsilon,C>0$. There is a set $X_{s,\varepsilon,C}$ of $N^{O_{s,\varepsilon,C}(1)}$ $s$-step nilsequences of complexity $O_{s,\varepsilon,C}(1)$ such that, for any $s$-step nilsequence $\phi$ of complexity at most $C$, there is some $\phi^{\prime}\in X_{s,\varepsilon,C}$ with $\left|\phi(d)-\phi^{\prime}(d)\right|\leqslant\varepsilon$ for $d=1,\ldots,N$.

Combining Conjecture 2.1 with Proposition 2.2 yields the following corollary.

Corollary 2.3**.**

Conjecture 2.1 holds if and only if it holds under the restriction that the cardinality of the set of 2-step nilsequences is $N^{O_{\varepsilon}(1)}$.

Theorem 2.4**.**

Let $S\subset[N]$ be formed by letting each $d\in[N]$ lie in $S$ independently with probability $\sigma:=\omega(\log N/N)$. If Conjecture 2.1 holds, then Szemerédi’s theorem for $k=3$ with common differences in $S$ holds with probability $1-o(N^{-100})$.

Proof.

For $d=1,\ldots,N$, let $Y(d)$ denote the mean-zero random variable $1_{S}(d)-\sigma$. We will show that, for any $\varepsilon>0$,

[TABLE]

Then, with probability $1-o(N^{-100})$, we have that

[TABLE]

uniformly in $A$. In particular, by Varnavides’ theorem [Var59], the average number of 3APs in $A$ (that is, $\mathbb{E}_{x,d}1_{A}(x)1_{A}(x+d)1_{A}(x+2d)$) is bounded away from zero (uniformly in $N$). Sending $\varepsilon\to 0$ yields that, with probability $1-o(N^{-100})$,

[TABLE]

uniformly in $A$.

It remains to show (2.1). To this end, let $\varepsilon>0$ and induce Conjecture 2.1 and Corollary 2.3 to write

[TABLE]

where the sum is over $N^{O_{\varepsilon}(1)}$ nilsequences of complexity $O_{\varepsilon}(1)$. Then,

[TABLE]

We will work on each of these terms separately.

Firstly, by the union bound, we have that

[TABLE]

where $\phi$ is some 2-step nilsequence of complexity $O_{\varepsilon}(1)$. In particular, $\phi$ is bounded in terms of $\varepsilon$; that is, $\left|\phi(d)\right|=O_{\varepsilon}(1)$ for $d\in[N]$. Note then that the $\phi(d)Y(d)$ are independent mean zero random variables with variance $O_{\varepsilon}(\sigma)$. By Bernstein’s inequality ([Ber46]),

[TABLE]

Combining this inequality with (2.3), we have

[TABLE]

since $\sigma=\omega(\log N/N)$.

For the error term, using again Bernstein’s inequality in the penultimate line, we have

[TABLE]

Now, combining (2.2), (2) and (2), we have that

[TABLE]

when $\sigma=\omega(\log N/N)$, completing the proof. ∎

Theorem 2.5**.**

Let $S\subset[N]$ be formed by letting each $d\in[N]$ lie in $S$ independently with probability $\mu(d)=\omega(1/d)$. If Conjecture 2.1 holds, then Szemerédi’s theorem for $k=3$ with common differences in $S$ holds with probability $1-o(N^{-100})$.

Proof.

The proof is essentially the same as that of Theorem 2.4. We will focus on points of the argument that differ.

We will show that for any $\varepsilon>0$,

[TABLE]

where this time $Y(d)=1_{S}(d)-\sigma$, $1_{S}(d)$ is Bernoulli with parameter $\mu(d)=\omega(1/d)$ and $\sigma=\mathbb{E}_{d}\mu(d)$. Having established (2.7), it is easily checked that the rest of the proof is identical with this slightly-different definition of $\sigma$.

The main difference in establishing (2.7) is that now the $Y(d)$ are not mean zero, and so we cannot conclude (2.4) immediately from Bernstein’s inequality. However, one checks that

[TABLE]

and so modifying to the left hand side of (2.4) before inducing Bernstein’s inequality yields a proof of (2.4). One also easily proves (2) for our newly-defined $Y(d)$. We omit the details; the rest of the argument remains the same. ∎

The following corollary extends the finitary result obtained in Theorem 2.5 to a result in $\mathbb{N}$. It says, in particular, that the conjectures we inherited from [Fra16, Problem 31] and [FLW16, Conjecture 2.5] are true for $k=3$ under our Conjecture 2.1.

Corollary 2.6**.**

Let $S\subset\mathbb{N}$ be chosen at random with $\mathbb{P}(n\in S)=\omega(1/n)$. Then, if Conjecture 2.1 holds, it is almost surely the case that all subsets of $\mathbb{N}$ with positive upper density contain a 3-term arithmetic progression with common difference in $S$.

Proof.

For $B\subset\mathbb{N}$, let $E_{B}$ be the event that $B$ contains a 3AP with common difference in $S$, and let $E_{B,N}$ be the event that $B\cap[N]$ contains a 3AP with common difference in $S\cap[N]$. Then $E_{B}=\bigcup_{N=1}^{\infty}E_{B,N}$.

For $m=1,2,\ldots$, let $I_{m}=\{B\subset\mathbb{N}:\limsup_{N}|B\cap[N]|/N\geqslant 1/m\}$, and for each $N$ let $I_{m,N}=\{B\subset\mathbb{N}:|B\cap[N]|/N\geqslant 1/m\}$. Observe that $I_{m}=\limsup_{N}I_{m,N}$.

Let $G_{m}$ be the event that all $B\in I_{m}$ contain a 3AP with common difference in $S$, that is $G_{m}=\cap_{B\in I_{m}}E_{B}$. Then the probability that Szemerédi’s theorem with common difference in $S$ holds is given by $\mathbb{P}\left(\bigcup_{m=1}^{\infty}G_{m}\right)$. By the monotone convergence theorem, this is equal to $\lim_{m\to\infty}\mathbb{P}(G_{m})$. We will show that $\mathbb{P}(G_{m}^{c})=0$ for all $m$. To this end we compute,

[TABLE]

From Theorem 2.5 it follows that $\mathbb{P}\left(\bigcup_{B\in I_{m,N}}E_{B,N}^{c}\right)=o(N^{-100})$, so that $\sum_{N=1}^{\infty}\mathbb{P}\left(\bigcup_{B\in I_{m,N}}E_{B,N}^{c}\right)<\infty.$ Thus, by the Borel–Cantelli lemma, we have that $\mathbb{P}\left(\limsup_{N}\bigcup_{B\in I_{m,N}}E_{B,N}^{c}\right)=0$. The result follows. ∎

3. Over finite fields

Recall that for 2-term arithmetic progressions, $\mathbb{P}(d\in S)=\omega(\log N/N)$ was sufficient for Szemerédi’s theorem with common difference in $S$ to hold asymptotically almost surely. It is not difficult to prove that the analogous fact is true over finite fields: that $\mathbb{P}(d\in S)=\omega(\log(p^{n})/p^{n})$ is sufficient. (One uses similar Fourier-analytic arguments to the $k=2$ case over the integers; here there is only a discrete set of additive characters so the argument is even easier.)

We showed in the previous section that, under Conjecture 2.1, if elements of $[N]$ are chosen to lie in $S$ with probability $\omega(\log|[N]|/|[N]|)$ then Szemerédi’s theorem for $k=3$ almost surely holds with common difference in $S$. In this section, we show that the analogous result over $\mathbb{F}_{p}^{n}$ is not true. In fact, if elements are selected to lie in $S$ independently with probability

[TABLE]

where $c=1/2-o(1)$, then there will almost surely exist a set $A$ with positive density such that $A$ contains no 3APs with common difference in $S$. (Actually we deal with a slightly different probability model for convenience, but the above statement is an easy consequence of Corollary 3.2.)

The reason for the different behavior is that there are far more quadratic obstructions to 3APs in $\mathbb{F}_{p}^{n}$. Indeed, for $M\in M_{n}(\mathbb{F}_{p})$, define $A_{M}=\{x:x^{\top}Mx=0\}$ and note that $A_{M}$ has positive density (uniformly in $n$). One observes that if $x,x+d,x+2d\in A_{M}$, then $d^{\top}Md=0$, that is, $d\in A_{M}$. It follows that if all $A_{M}$ are to have 3APs with common difference in $S$, then $S$ must have the following property: for all $M\in M_{n}(\mathbb{F}_{p})$, there exists $d\in S$ such that $d^{\top}Md=0$.

Theorem 3.1**.**

Fix $p$ an odd prime. If $S\subset\mathbb{F}_{p}^{n}$ is formed by selecting at most $\binom{n+1}{2}-11n\log_{p}n$ elements of $\mathbb{F}_{p}^{n}$ independently at random, then almost surely as $n\to\infty$ there exists some $M\in M_{n}(\mathbb{F}_{p})$ such that $d^{\top}Md\neq 0$ for all $d\in S$.

Corollary 3.2**.**

Fix $p$ an odd prime. If $S\subset\mathbb{F}_{p}^{n}$ is formed by selecting at most $\binom{n+1}{2}-11n\log_{p}n$ elements of $\mathbb{F}_{p}^{n}$ independently at random, then almost surely as $n\to\infty$ there exists some set $A\subset\mathbb{F}_{p}^{n}$ of positive density such that $A$ contains no 3-term arithmetic progression with common difference in $S$.

The remainder of this section will prove Theorem 3.1.

It suffices to consider symmetric matrices because if $M\in M_{n}(\mathbb{F}_{p})$ then, letting $M^{\prime}=(M+M^{\top})/2\in S_{n}(\mathbb{F}_{p})$, we have $d^{\top}Md=d^{\top}M^{\prime}d$ for all $d$. We identify $S_{n}(\mathbb{F}_{p})$ with $\mathbb{F}_{p}^{\binom{n+1}{2}}$ naturally; we will write $M_{v}$ for the matrix corresponding to a vector $v$, and $v_{M}$ for the vector corresponding to a matrix $M$. Define $\varphi:\mathbb{F}_{p}^{n}\to\mathbb{F}_{p}^{{\binom{n+1}{2}}}$ to be the degree 2 Veronese map, that is $(d_{1},\ldots,d_{n})\mapsto(d_{i}d_{j})_{1\leqslant i\leqslant j\leqslant n}$. Then $d^{\top}Md=\varphi(d)\cdot v_{M}$ and so $d^{\top}Md\neq 0$ for all $d\in S$ if and only if $v_{M}\not\in\cup_{d\in S}\varphi(d)^{\perp}$.

The following lemma demonstrates that if $\varphi(S)$ is linearly independent then there exists some matrix $M$ with $v_{M}\not\in\cup_{d\in S}\varphi(d)^{\perp}$.

Lemma 3.3**.**

Let $\{v_{1},\ldots,v_{k}\}$ be linearly independent in an $m$-dimensional vector space over $\mathbb{F}_{p}$. Then

[TABLE]

In particular,

[TABLE]

Proof.

Linear algebra; we omit the details. ∎

The goal will now be to show that, almost surely as $n\to\infty$, the elements $\varphi(d),d\in S$ are linearly independent. Let $\mathcal{W}_{k}$ be the set of all $k$ dimensional subspaces of $\mathbb{F}_{p}^{\binom{n+1}{2}}$ and let $W_{k}\in\mathcal{W}_{k}$ be a subspace such that $|W_{k}\cap\operatorname{Im}\varphi|=\max_{W\in\mathcal{W}_{k}}|W\cap\operatorname{Im}\varphi|$.

Lemma 3.4**.**

Let $N=|S|$. The probability that $\varphi(S)$ is linearly independent is bounded below by

[TABLE]

Proof.

Sampling $S$ by selecting elements $d_{1},\ldots,d_{N}$ at independently at random, the probability that $\varphi(S)$ is linearly independent is bounded below by,

[TABLE]

∎

By Lemma 3.4, to show that $\varphi(S)$ is almost surely linearly independent, it suffices now to show that $\mathbb{P}_{d\in\mathbb{F}_{p}^{n}}\left(\varphi(d)\in W_{N}\right)=o(1/n^{2})$. As an intermediate step, we will show in Proposition 3.7 that

[TABLE]

We separate out the main analytic observations in the following two lemmas. The first follows from orthogonality of characters.

Lemma 3.5**.**

Let $V$ be a vector space of functions $\mathbb{F}_{p}^{k}\to\mathbb{F}_{p}$ under pointwise operations. Let $\omega=\exp\left(2\pi i/p\right)$. Say $V(x)=0$ if $v(x)=0$ for all $v\in V$. Then

[TABLE]

The second is a standard estimate for Gauss sums.

Lemma 3.6**.**

Let $M$ be a symmetric matrix over $\mathbb{F}_{p}$. Then,

[TABLE]

Proof.

The result follows from taking square roots after the following computation:

[TABLE]

∎

Proposition 3.7**.**

We can bound $\mathbb{P}_{d}(\varphi(d)\in W_{N})$ as follows:

[TABLE]

where the expectation is taken over all $M_{v}:v\in W_{N}^{\perp}$.

Proof.

Using Lemma 3.5 (with $V$ the vector space of quadratic forms defined by $\{M_{v}\}$) and Lemma 3.6 we can compute that

[TABLE]

∎

Finally, it remains to show that

[TABLE]

We do so with the following (crude) observations. Firstly, the number of $n\times n$ matrices of rank at most $r$ is bounded above by $p^{2nr}$ (choose the row space in at most $p^{nr}$ ways and then choose each of the $n$ rows in at most $p^{r}$ ways). Then, splitting the sum by rank, we have

[TABLE]

and so, recalling that $\left|\{M_{v}\}\right|=p^{\binom{n+1}{2}-N}=p^{11n\log_{p}n}$,

[TABLE]

This completes the proof of Theorem 3.1.

Appendix A Some points on nilsequences

We will briefly recall the main objects associated with nilsequences. Our use of the term nilsequence essentially coincides with the definition of ‘polynomial nilsequence’ in [GTZ12, Definition 4.1]. We direct an interested or concerned reader there for properly developed definitions and details.

The following definitions are essentially consistent with [GTZ12, Definition 4.1]. Throughout this section let $G$ be a connected, simply-connected, nilpotent Lie group with Lie algebra $\mathfrak{g}$. Let $\Gamma$ be a lattice (discrete, cocompact subgroup) in $G$, whence $G/\Gamma$ is a nilmanifold. Let $G_{\bullet}=(G_{i})_{i=0}^{s+1}$ be an $s$-step Lie filtration which is rational with respect to $\Gamma$ in the sense that $\Gamma\cap G_{i}$ is a lattice in $G_{i}$ for all $i$. Let $p:[N]\to G$ be a polynomial sequence with respect to $G_{\bullet}$. Let $\psi$ be a Lipschitz continuous function $G\to\mathbb{C}$ which is $\Gamma$-automorphic (we will often abuse notation and consider $\psi$ as a function on $G/\Gamma$). With this setup, defining $\phi(n):=\psi(p(n)\Gamma)$ yields a nilsequence.

We will also make a couple of minor amendments to [GTZ12, Definition 4.1]. Firstly, we will also add to the data associated to a nilsequence a Mal’cev basis $\mathscr{B}$ for the Lie algebra, which is adapted to the Lie filtration of $G$. This, in particular, is a basis with respect to which the structure constants of the Lie algebra $\mathfrak{g}$ are rational (the existence of such a basis is due to Mal’cev [Mal49]). See [GT12, Chapter 2] and in particular [GT12, Definition 2.1] and the remarks that follow it for more details and discussion. Also, rather than use the left-invariant Riemannian metric on $G/\Gamma$ as in [GTZ12, Definition 4.1], we will borrow the right-invariant metric $d_{\mathscr{B}}=d$ on $G/\Gamma$ from [GT12, Definition 2.2].

We adopt a similar notion of the complexity of a nilsequence to that used in the formulation of the inverse conjecture for the Gowers $U^{s+1}[N]$ norm in [GTZ12, Conjecture 4.5]. When we refer to a nilsequence $\phi$ as having complexity bounded by $C>0$, we take as part of the definition that the following are also bounded by $C$:

• •

the dimension of $G$,

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the heights of the (rational) structure constants of the Lie bracket operation with respect to $\mathscr{B}$,

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the heights of the (rational) coordinates of $\log(g_{\Gamma})$ with respect to $\mathscr{B}$ for all $g_{\Gamma}\in S_{\Gamma}$, where $S_{\Gamma}$ is some generating set for $\Gamma$,

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the Lipschitz constant of $\psi$ (with respect to the metric $d$ in the domain), and

• •

$\|\psi\|_{\infty}$.

The goal of the remainder of the appendix is to prove Proposition 2.2. Our first lemma is standard and follows from the definitions above.

Lemma A.1**.**

Let $C>0$ and let $\phi$ be an $s$-step nilsequence of complexity at most $C$. Then there are $O_{s,C}(1)$ possibilities for the Lie filtration $G_{\bullet}$ associated to $\phi$, and $O_{C}(1)$ possibilities for the lattice $\Gamma$.

The following lemma is the key ingredient in the proof of Proposition 2.2.

Lemma A.2** ([BGSZ16, Lemma B.7]).**

Let $G/\Gamma$ be an $s$-step nilmanifold and let $\varepsilon\in(0,1/2)$. There exists a set $P$ of $N^{O_{s,\varepsilon}(1)}$ polynomial sequences $p^{\prime}:\mathbb{Z}\to G$ such that for every polynomial sequence $p$ in $G$ there exists $p^{\prime}\in P$ with $d(p(n)\Gamma,p^{\prime}(n)\Gamma)<\varepsilon$ for all $n\in[N]$.

The final ingredient is a consequence of the Arzelà–Ascoli theorem, or can be verified from first principles.

Lemma A.3**.**

Let $\varepsilon>0$ and let $G/\Gamma$ be a nilmanifold associated to a nilsequence as above. Let $\mathcal{F}$ be the family of Lipschitz functions $G/\Gamma\to\mathbb{C}$ which are bounded by $C$ and have Lipschitz constant at most $C$. Then there exists a constant $K_{\varepsilon,C}$ and a set $\mathcal{F}^{\prime}$ of cardinality $O_{\varepsilon,C}(1)$ containing Lipschitz functions $G/\Gamma\to\mathbb{C}$ which are bounded by $C$ and have Lipschitz constant at most $K_{\varepsilon,C}$ with the following property. For every $\psi\in\mathcal{F}$, there exists $\psi^{\prime}\in\mathcal{F}^{\prime}$ with $|\psi(g)-\psi^{\prime}(g)|<\varepsilon$ for all $g\in G/\Gamma$.

Proof of Proposition 2.2.

By Lemma A.1, there are $O_{s,C}(1)$ Lie groups, Lie filtrations and lattices corresponding to nilsequences of complexity at most $C$. Henceforth fix a particular nilmanifold $G/\Gamma$. Let $\phi(n)=\psi(p(n)\Gamma)$ be a nilsequence of complexity at most $C$ with nilmanifold $G/\Gamma$. Define $p^{\prime}$ to be the polynomial sequence in $G$ produced by Lemma A.2 with parameter $\frac{\varepsilon}{2C}$. Also, define $\psi^{\prime}$ to be the Lipschitz function $G/\Gamma\to\mathbb{C}$ produced by Lemma A.3 with parameter $\varepsilon/2$. Define the nilsequence $\phi^{\prime}$ by $\phi^{\prime}(n)=\psi^{\prime}(p^{\prime}(n)\Gamma)$. It follows that,

[TABLE]

for all $n\in[N]$. Since we have chosen from a family of $N^{O_{s,\varepsilon,C}(1)}$ such polynomial sequences $p^{\prime}$ and $O_{\varepsilon,C}(1)$ such Lipschitz functions $\psi^{\prime}$, we have chosen $\phi^{\prime}$ from a family of cardinality $N^{O_{s,\varepsilon,C}(1)}$. It is also clear that these nilsequences have complexity $O_{s,\varepsilon,C}(1)$. The result follows. ∎

Acknowledgments

The author is indebted to his supervisor Ben Green for his guidance and support on this project.