Minimum number of additive tuples in groups of prime order

TL;DR

This paper investigates the minimum number of additive tuples in subsets of prime order groups, revealing extremal configurations and their properties using Fourier analysis, especially when the tuple size parameter varies.

Contribution

It extends previous results by analyzing the structure of extremal configurations for large tuple sizes in prime order groups, especially when the tuple size parameter is congruent to 1 modulo p.

Findings

Extremal configurations are intervals for certain parameters.

When k ≡ 1 mod p and p ≥ 13, extremal sets alternate between at least two affine non-equivalent forms.

Fourier analysis reveals structural properties of these extremal sets.

Abstract

For a prime number $p$ and a sequence of integers $a_0,\dots,a_k\in \{0,1,\dots,p\}$, let $s(a_0,\dots,a_k)$ be the minimum number of $(k+1)$-tuples $(x_0,\dots,x_k)\in A_0\times\dots\times A_k$ with $x_0=x_1+\dots + x_k$, over subsets $A_0,\dots,A_k\subseteq\mathbb{Z}_p$ of sizes $a_0,\dots,a_k$ respectively. An elegant argument of Lev (independently rediscovered by Samotij and Sudakov) shows that there exists an extremal configuration with all sets $A_i$ being intervals of appropriate length, and that the same conclusion also holds for the related problem, reposed by Bajnok, when $a_0=\dots=a_k=:a$ and $A_0=\dots=A_k$, provided $k$ is not equal 1 modulo $p$. By applying basic Fourier analysis, we show for Bajnok's problem that if $p\ge 13$ and $a\in\{3,\dots,p-3\}$ are fixed while $k\equiv 1\pmod p$ tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.