Arithmetic Progressions of Length Three in Multiplicative Subgroups of $\mathbb{F}_p$

TL;DR

This paper presents an efficient algorithm for detecting non-trivial three-term arithmetic progressions in multiplicative subgroups of finite fields, enabling polynomial-time computation of related combinatorial numbers.

Contribution

The paper introduces a significantly more efficient algorithm for finding 3-APs in multiplicative subgroups, improving over naive methods.

Findings

Algorithm detects 3-APs more efficiently

Polynomial-time computation of Var der Waerden-like numbers

Enhanced understanding of arithmetic progressions in finite fields

Abstract

In this paper, we give an algorithm for detecting non-trivial 3-APs in multiplicative subgroups of $\mathbb{F}_p^\times$ that is substantially more efficient than the naive approach. It follows that certain Var der Waerden-like numbers can be computed in polynomial time.

Equations

$$\frac{1}{p}\sum _{k=0}^{p-1}{e}^{\frac{-2\pi ik}{p}x}=\{\begin{array}{ll}1,& \text{ if }x\equiv 0(\mathrm{mod}p);\\ 0,& \text{ if }x\not\equiv 0(\mathrm{mod}p).\end{array}$$

$$\mathcal{N}=\sum _{x\in A}\sum _{y\in A}\sum _{z\in A}\frac{1}{p}\sum _{k=0}^{p-1}{e}^{\frac{-2\pi ik}{p}(x+y-2z)}$$

$$\frac{1}{p}\sum _{k=0}^{p-1}\sum _{x\in A}\sum _{y\in A}\sum _{z\in A}{e}^{\frac{-2\pi ik}{p}x}\cdot e^{\frac{-2\pi ik}{p}y}\cdot e^{\frac{2\pi ik}{p}z}$$

$$=\frac{1}{p}\sum _{k=0}^{p-1}\left[\sum _{x\in A}{e}^{\frac{-2\pi ik}{p}x}\cdot \sum _{y\in A}{e}^{\frac{-2\pi ik}{p}y}\cdot \sum _{z\in A}{e}^{\frac{2\pi ik}{p}2z}\right]$$

$$=\frac{1}{p}\sum _{k=0}^{p-1}\left[\sum _{x\in {\mathbb{F}}_p}{\mathrm{Ch}}_A(x)e^{\frac{-2\pi ik}{p}x}\cdot \sum _{y\in {\mathbb{F}}_p}{\mathrm{Ch}}_A(y)e^{\frac{-2\pi ik}{p}y}\cdot \sum _{z\in {\mathbb{F}}_p}{\mathrm{Ch}}_A(-2z)e^{\frac{2\pi ik}{p}z}\right]$$

$$=\frac{1}{p}\sum _{k=0}^{p-1}{\widehat{\mathrm{Ch}}}_A(k{)}^2\cdot {\widehat{\mathrm{Ch}}}_A(-2k),$$

$$\hat{f}(x)=\sum _{k=0}^{p-1}f(k)e^{\frac{-2\pi ik}{p}x}.$$

$$\text{eqref}eq3$$

$$=\frac{|A{|}^3}{p}+\frac{1}{p}\sum _{k=1}^{p-1}{\widehat{\mathrm{Ch}}}_A(k{)}^2\cdot {\widehat{\mathrm{Ch}}}_A(-2k)$$

$$={\delta }^3{p}^2+\frac{1}{p}\sum _{k=1}^{p-1}{\widehat{\mathrm{Ch}}}_A(k{)}^2\cdot {\widehat{\mathrm{Ch}}}_A(-2k).$$

$$\left|\frac{1}{p}\sum _{k=1}^{p-1}{\widehat{\mathrm{Ch}}}_A(k{)}^2\cdot {\widehat{\mathrm{Ch}}}_A(-2k)\right|$$

$$\le \alpha \left|\sum _{k=1}^{p-1}{\widehat{\mathrm{Ch}}}_A(k{)}^2\right|$$

$$\le \alpha p\left|\sum _{k=1}^{p-1}{\mathrm{Ch}}_A(k{)}^2\right|$$

$$\le \alpha \delta p^2.$$

$${\delta }^3{p}^2>\delta p+\alpha \delta p^2.$$

$${\delta }^3{p}^2\ge \delta p+p^{-1/2}\delta p^2$$

$$⟺{\delta }^{2}p\ge 1+p^{1/2}$$

$$⟺(p-1{)}^2\ge n^{2}p(1+p^{1/2})$$

$$(aH+bH)\cap cH\ne \varnothing \text{ if and only if }(c-aH)\cap bH\ne \varnothing .$$

$$\sum _{\begin{array}{c}p\le (1+\epsilon )n^4\\ p\equiv 1(\mathrm{mod}n)\end{array}}\mathcal{O}(p)=\mathcal{O}\left(\sum _{\begin{array}{c}p\le (1+\epsilon )n^4\\ p\equiv 1(\mathrm{mod}n)\end{array}}p\right).$$

$$\sum _{\begin{array}{c}p\le x\\ p\equiv 1(\mathrm{mod}n)\end{array}}p$$

$$\mathcal{O}\left(\sum _{\begin{array}{c}p\le (1+\epsilon )n^4\\ p\equiv 1(\mathrm{mod}n)\end{array}}p\right)$$

$$=\mathcal{O}\left(\frac{n^8}{\log (n)}\right)$$

Full text

1 Introduction

Additive structures inside multiplicative subgroups of $\mathbb{F}_{p}^{\times}$ have recently received attention. Alon and Bourgain [1] study solutions to $x+y=z$ in $H<\mathbb{F}_{p}^{\times}$, and Chang [2] studies arithmetic progressions in $H<\mathbb{F}_{p}^{\times}$. In this paper, we define a Van der Waerden-like number for $H<\mathbb{F}_{p}^{\times}$ of index $n$, and give a polynomial-time algorithm for determining such numbers.

Definition 1**.**

Let $VW_{3}^{\times}(n)$ denote the least prime $q\equiv 1\pmod{n}$ such that for all primes $p\equiv 1\pmod{n}$ with $p\geq q$, the multiplicative subgroup of $\mathbb{F}_{p}^{\times}$ of index $n$ contains a mod-$p$ arithmetic progression of length three.

Our main results are the following two theorems:

Theorem 2**.**

$VW_{3}^{\times}(n)\leq(1+\varepsilon)n^{4}$* for all sufficiently large $n$ (depending on $\varepsilon$). In particular, $VW_{3}^{\times}(n)\leq 1.001n^{4}$ for all $n\geq 45$.*

Theorem 3**.**

$VW_{3}^{\times}(n)$* can be determined by an algorithm that runs in $\mathcal{O}(\frac{n^{8}}{\log n})$ time.*

Chang [2] proves that if $H<\mathbb{F}_{p}^{\times}$ and $|H|>cp^{3/4}$, then $H$ contains non-trivial 3-progressions. This implies our Theorem 2 with $(1+\varepsilon)n^{4}$ replaced by $cn^{4}$. We prove our Theorem 2 because we need to make the constant explicit.

2 Proof of Theorem 2

Proof.

We use one of the basic ideas of the proof of Roth’s Theorem on 3-progressions [3]. Let $A\subseteq\mathbb{F}_{p}$ with $|A|=\delta p$. Note that a 3-progression is a solution inside $A$ to the equation $x+y=2z$. Let $\mathcal{N}$ be the number of (possibly trivial) solutions to $x+y=2z$ inside $A$. We have that

[TABLE]

Because of (1), we have

[TABLE]

Rearranging (2), we get

[TABLE]

where $\operatorname{Ch}_{A}$ denotes the characteristic function of $A$, and $\hat{f}$ denotes the Fourier transform of $f$,

[TABLE]

Now we can pull out the $k=0$ term from (3):

[TABLE]

Let’s call $\delta^{3}p^{2}$ the main term, and $\frac{1}{p}\sum_{k=1}^{p-1}\hat{\operatorname{Ch}}_{A}(k)^{2}\cdot\hat{\operatorname{Ch}}_{A}(-k)$ the error term. We now bound this error term.

Suppose $0<\alpha<1$ and $|\hat{\operatorname{Ch}}_{A}(k)|\leq\alpha p$ for all $0\neq k\in\mathbb{F}_{p}$. In this case, we say that $A$ is $\alpha$-uniform. Then

[TABLE]

Therefore $\mathcal{N}\geq\delta^{3}p^{2}-\alpha\delta p^{2}$. Subtracting off the trivial solutions gives $\mathcal{N}-\delta p\geq\delta^{3}p^{2}-\delta p-\alpha\delta p^{2}$. Hence there is at least one non-trivial solution if

[TABLE]

Let $A=H$ be a multiplicative subgroup of $\mathbb{F}_{p}$ of index $n$. As is well-known (see for example [4, Corollary 2.5]), if $H$ is a multiplicative subgroup of $\mathbb{F}_{p}^{\times}$, then $H$ is $\alpha$-uniform for $\alpha\leq p^{-1/2}$. Thus it suffices to have

[TABLE]

where the last line follows from $\delta=(p-1)/(np)$. It is straightforward to check that (6) is satisfied by $p=(1+\varepsilon)n^{4}$ for sufficiently large $n$.

∎

The data gathered for $VW^{\times}_{3}(n)$, $n\leq 100$, suggest that the exponent of 4 on $n$ is too large; see Figure 1. These data are available at www.oeis.org, sequence number A298566.

3 A More General Framework

Before we establish our algorithm, it will helpful to generalize to arbitrary linear equations in three variables over $\mathbb{F}_{p}$. Suppose we’re looking for solutions to $ax+by=cz$ in $H<\mathbb{F}_{p}^{\times}$, for fixed $a,b,c\in\mathbb{F}_{p}^{\times}$. There is a solution just in case $(aH+bH)\cap cH$ is nonempty.

The following result affords an algorithmic speedup in counting solutions to $ax+by=cz$ inside $H$:

Lemma 4**.**

For $a$, $b$, $c\in\mathbb{F}_{p}^{\times}$ and $H<\mathbb{F}_{p}^{\times}$,

[TABLE]

Notice that while the implied computation on the left side of the biconditional is $\mathcal{O}(p^{2})$, the one on the right is $\mathcal{O}(p)$, since we compute $|H|$ subtractions and $|H|$ comparisons. (We consider the index $n$ fixed.)

Proof.

Let $H=\{g^{kn}:0\leq k<(p-1)/n\}$, where $n$ is the index of $H$ and $g$ is a primitive root modulo $p$. Fix $a,b,c\in\mathbb{F}_{p}$.

For the forward direction, suppose $(aH+bH)\cap cH\neq\varnothing$, so there are $x,y,z\in H$ such that $ax+by=cz$. Then $by=cz-ax$. Multiplying by $z^{-1}\in H$ yields $b(yz^{-1})=c-a(xz^{-1})$. Therefore $(c-aH)\cap bH\neq\varnothing$. The other direction is similar. ∎

Lemma 4 allows us to detect solutions to linear equations in linear time. The caveat for the case $a=b=1$, $c=2$ is that $H+H$ always contains $2H$, since $h+h=2h$ for all $h\in H$; these solutions correspond to the trivial 3-APs $h,h,h$. (Similarly, $(2-H)\cap H$ is always nonempty, since $1\in H$ and $2-1=1$.) To account for this, we simply consider $H^{\prime}$ = $H\setminus\{1\}$, and calculate $(2-H^{\prime})\cap H^{\prime}$ instead.

4 Proof of Theorem 3

Proof.

Here is the algorithm.

We now argue that Algorithm 1 runs in $\mathcal{O}\left(\frac{n^{8}}{\log n}\right)$ time. Since calculating $(2-H^{\prime})\cap H^{\prime}$ is $\mathcal{O}(p)$ for each prime $p$, our runtime is bounded by

[TABLE]

A standard estimate on the prime sum

[TABLE]

is asymptotically $\frac{x^{2}}{\varphi(n)\log x}$, giving

[TABLE]

as desired. ∎

Our timing data suggest that the correct runtime might be more like $\mathcal{O}(n^{6})$; see Figure 2.

5 Further Directions

For any $a,b,c\in\mathbb{Z}^{+}$, we can define an analog to $VW^{\times}_{3}(n)$ by considering the equation $ax+by=cz$ instead of $x+y=2z$. (Assume $p$ is greater than $a$, $b$, and $c$.) The bound from Theorem 2 stays the same if $a+b=c$ and goes down to $n^{4}+5$ otherwise. But as suggested by the data in Figure 1, these bounds are not tight. How does the choice of $a$, $b$, and $c$ affect the growth rate of the corresponding Van der Waerden-like number? Clearly $VW^{\times}_{3}(n)$ is not monotonic, but it appears to bounce above and below some “average” polynomial growth rate. Will that growth rate vary with the choice of $a$, $b$, and $c$? Does it depend on whether $a+b=c$ only?

6 Acknowledgements

The author wishes to thank Andrew Shallue and Valentin Andreev for many productive conversations.