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Difference sets and positive exponential sums II: cubic residues in cyclic groups

Mate Matolcsi, Imre Z. Ruzsa

TL;DR

The paper addresses bounding the maximal density of a subset $B_q$ of the cyclic group $\mathbb{Z}_q$ whose difference set avoids cubic residues, by constructing nonnegative exponential sums in the form of modular witnesses $g^{(q)}$ and their corresponding $k$th-power witnesses $f^{(n)}$. It develops a quantitative framework using the parameters $\delta(C_0^{(q)})$ and $\lambda(C_0^{(q)})$ and introduces self-compatibility to link modular constructions across moduli. For squarefree moduli $q$, it obtains explicit upper bounds on $\lambda(C_0^{(q)})$ via prime-factor analysis, and then extends to general $q$ using prime-power recurrences and direct products, culminating in a self-compatible family with $\lambda(C_0^{(q)})\le q^{-\varepsilon}$ for $\varepsilon\approx 0.1195$, which yields $\delta(B_q) \le q^{-\varepsilon}$ for all $q$. These modular results illuminate the path toward strong integer-case bounds on sets with cubes-avoiding differences and set the stage for future work on higher powers and broader moduli.

Abstract

By constructing suitable nonnegative exponential sums we give upper bounds on the cardinality of any set $B_q$ in cyclic groups $\ZZ_q$ such that the difference set $B_q-B_q$ avoids cubic residues modulo $q$.

Difference sets and positive exponential sums II: cubic residues in cyclic groups

TL;DR

The paper addresses bounding the maximal density of a subset of the cyclic group whose difference set avoids cubic residues, by constructing nonnegative exponential sums in the form of modular witnesses and their corresponding th-power witnesses . It develops a quantitative framework using the parameters and and introduces self-compatibility to link modular constructions across moduli. For squarefree moduli , it obtains explicit upper bounds on via prime-factor analysis, and then extends to general using prime-power recurrences and direct products, culminating in a self-compatible family with for , which yields for all . These modular results illuminate the path toward strong integer-case bounds on sets with cubes-avoiding differences and set the stage for future work on higher powers and broader moduli.

Abstract

By constructing suitable nonnegative exponential sums we give upper bounds on the cardinality of any set in cyclic groups such that the difference set avoids cubic residues modulo .
Paper Structure (2 sections, 5 theorems, 18 equations)

This paper contains 2 sections, 5 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Cubic residues

Key Result

Theorem 2.1

Let $q=p_1p_2\dots p_rs_1s_2\dots s_m$ be a squarefree integer, where $p_i$ denote primes of the form $3k+2$, $s_l$ denote primes of the form $3k+1$, and if $3|q$ then we list the prime 3 among the $p_i$. There exists a cubic witness function modulo $q$, $g^{(q)}(y)=b_0+ \sum_{j=1}^{q-1} b_j (e(j^3 That is, with the notation of Definition ldef, we have for every $\varepsilon >0$.

Theorems & Definitions (12)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • ...and 2 more