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Shifted multiplicative subgroups are not ratio sets

Seoyoung Kim, Chi Hoi Yip, Semin Yoo

Abstract

In a recent breakthrough, Kalmynin proved a conjecture of Lev--Sonn and a conjecture of Sárközy on additive decompositions of multiplicative subgroups of a prime field. In this paper, we prove a multiplicative analogue of Kalmynin's result on a generalization of the Lev--Sonn conjecture, inspired by a relevant conjecture of Sárközy. We show that all nonzero shifts of proper multiplicative subgroups (of size at least $3$) are not ratio sets of the form $A/A$. This in particular extends a result of Shkredov, where he showed the same for small multiplicative subgroups (of size $<p^{6/7}$ in $\mathbb{F}_p$). We also prove an analogous statement over complex numbers for finite subgroups of the unit circle, which may be of independent interest.

Shifted multiplicative subgroups are not ratio sets

Abstract

In a recent breakthrough, Kalmynin proved a conjecture of Lev--Sonn and a conjecture of Sárközy on additive decompositions of multiplicative subgroups of a prime field. In this paper, we prove a multiplicative analogue of Kalmynin's result on a generalization of the Lev--Sonn conjecture, inspired by a relevant conjecture of Sárközy. We show that all nonzero shifts of proper multiplicative subgroups (of size at least ) are not ratio sets of the form . This in particular extends a result of Shkredov, where he showed the same for small multiplicative subgroups (of size in ). We also prove an analogous statement over complex numbers for finite subgroups of the unit circle, which may be of independent interest.
Paper Structure (8 sections, 7 theorems, 72 equations)

This paper contains 8 sections, 7 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric polynomials
  4. A multiplicative analogue of Hanson--Petridis polynomials
  5. Proof of \ref{['thm: mul version of Lev-Sonn']}
  6. Proof of \ref{['thm: mul version of Lev-Sonn']}(1)
  7. Proof of \ref{['thm: mul version of Lev-Sonn']}(2)
  8. The complex setting

Key Result

Theorem 1.3

Let $p$ be a prime and $G$ be a proper multiplicative subgroup of ${\mathbb F}_p$. If $A,B$ are subsets of ${\mathbb F}_p$ such that $A+B \subseteq G \cup \{0\}$, then

Theorems & Definitions (19)

  • Conjecture 1.1: Sárközy S12
  • Conjecture 1.2
  • Theorem 1.3: Hanson and Petridis HP
  • Conjecture 1.4: Lev and Sonn LS17
  • Theorem 1.5: Kalmynin Kalmynin
  • Conjecture 1.6: Sárközy S14
  • Conjecture 1.7: KYY
  • Lemma 1.8: KYY
  • Theorem 1.9
  • Theorem 1.10
  • ...and 9 more