# On a question of Sidorenko

## TL;DR

This paper investigates a conjecture by Sidorenko regarding the maximum number of functions with certain bijective difference properties over modular integers, providing counterexamples for specific composite numbers.

## Contribution

The paper disproves Sidorenko's conjecture for n=15, 21, and 27 by constructing explicit counterexamples using computational methods.

## Key findings

- Counterexamples for n=15, 21, 27 found
- Sidorenko's conjecture does not hold for these n
- The maximum number of such functions exceeds the minimal prime divisor in these cases

## 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.