On a question of Sidorenko

Abstract

For a positive integer $n>1$ denote by $\omega(n)$ the maximal possible number $k$ of different functions $f_1,\dots,f_k:\mathbb{Z}/n\mathbb{Z}\mapsto \mathbb{Z}/n\mathbb{Z}$ such that each function $f_i-f_j,i<j$, is bijective. Recently A. Sidorenko conjectured that $\omega(n)$ equals to the minimal prime divisor of $n$. We disprove it for $n=15,21,27$ by several counterexamples found by computer.