On a question of Sidorenko

D. Cherkashin; F. Petrov; V. Sokolov

arXiv:1901.00440·math.CO·January 3, 2019

On a question of Sidorenko

D. Cherkashin, F. Petrov, V. Sokolov

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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.

Findings

01

Counterexamples for n=15, 21, 27 found

02

Sidorenko's conjecture does not hold for these n

03

The maximum number of such functions exceeds the minimal prime divisor in these cases

Abstract

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

Equations18

\begin{array}[]{ccccccccccccccc}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&1&2&3&4&5&6&7&8&9&10&11&12&13&14\\ 0&9&3&2&13&11&10&12&4&6&8&14&7&5&1\\ 0&12&4&11&10&9&5&2&6&14&7&3&13&1&8\end{array}

\begin{array}[]{ccccccccccccccc}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&1&2&3&4&5&6&7&8&9&10&11&12&13&14\\ 0&9&3&2&13&11&10&12&4&6&8&14&7&5&1\\ 0&12&4&11&10&9&5&2&6&14&7&3&13&1&8\end{array}

\begin{array}[]{ccccccccccccccccccccc}13&11&14&0&2&1&5&7&3&10&15&17&16&20&4&18&9&19&12&6&8\\ 14&5&4&13&9&18&2&15&6&10&17&1&11&19&8&3&7&12&0&16&20\end{array}

\begin{array}[]{ccccccccccccccccccccc}13&11&14&0&2&1&5&7&3&10&15&17&16&20&4&18&9&19&12&6&8\\ 14&5&4&13&9&18&2&15&6&10&17&1&11&19&8&3&7&12&0&16&20\end{array}

\begin{array}[]{ccccccccccccccccccccccccccc}12&17&11&20&5&19&1&9&0&13&15&18&6&10&22&3&2&8&14&25&4&24&21&16&7&23&26\\ 4&6&5&15&19&18&3&13&24&16&20&1&7&0&8&11&9&17&26&21&2&12&14&22&25&23&10\end{array}

\begin{array}[]{ccccccccccccccccccccccccccc}12&17&11&20&5&19&1&9&0&13&15&18&6&10&22&3&2&8&14&25&4&24&21&16&7&23&26\\ 4&6&5&15&19&18&3&13&24&16&20&1&7&0&8&11&9&17&26&21&2&12&14&22&25&23&10\end{array}

ω (nm) \geq min (ω (n), ω (m)) .

ω (nm) \geq min (ω (n), ω (m)) .

q_{t} (im + j) := f_{t} (i) m + g_{t} (j) .

q_{t} (im + j) := f_{t} (i) m + g_{t} (j) .

q_{t_{1}} (a) - q_{t_{2}} (a) = q_{t_{1}} (b) - q_{t_{2}} (b) \mbox m o d nm .

q_{t_{1}} (a) - q_{t_{2}} (a) = q_{t_{1}} (b) - q_{t_{2}} (b) \mbox m o d nm .

f_{t_{1}} (i) m + g_{t_{1}} (j) - f_{t_{2}} (i) m - g_{t_{2}} (j) = f_{t_{1}} (r) m + g_{t_{1}} (s) - f_{t_{2}} (r) m - g_{t_{2}} (s) \mbox m o d nm .

f_{t_{1}} (i) m + g_{t_{1}} (j) - f_{t_{2}} (i) m - g_{t_{2}} (j) = f_{t_{1}} (r) m + g_{t_{1}} (s) - f_{t_{2}} (r) m - g_{t_{2}} (s) \mbox m o d nm .

g_{t_{1}} (j) - g_{t_{2}} (j) = g_{t_{1}} (s) - g_{t_{2}} (s) \mbox m o d m,

g_{t_{1}} (j) - g_{t_{2}} (j) = g_{t_{1}} (s) - g_{t_{2}} (s) \mbox m o d m,

f_{t_{1}} (i) - f_{t_{2}} (i) = f_{t_{1}} (r) - f_{t_{2}} (r) \mbox m o d n .

f_{t_{1}} (i) - f_{t_{2}} (i) = f_{t_{1}} (r) - f_{t_{2}} (r) \mbox m o d n .

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TopicsAnalytic Number Theory Research · Advanced Combinatorial Mathematics · Advanced Mathematical Identities

Full text

On a question of Sidorenko

D. Cherkashina,c, F. Petrova,b, V. Sokolovb

†† a) St. Petersburg State University; b) St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences; c) St. Petersburg Branch of Higher School of Economics. E-mails: [email protected], [email protected],[email protected]

In a very recent paper [2] Sidorenko stated the following problem:

Let $G_{k}$ be a graph whose vertices are functions $f:\mathbb{Z}_{k}\to\mathbb{Z}_{k}$ . A pair of vertices $\{f,g\}$ forms an edge in $G_{k}$ if $f-g$ is a bijection. Lemma 2 restates the fact that $G_{k}$ has no triangles when $k$ is even. For odd $k$ , the problem of counting triangles in $G_{k}$ has been solved asymptotically in [1]. Let $p(k)$ be the smallest prime factor of $k$ . The $p(k)$ functions $f_{0},f_{1},\dots,f_{p(k)-1}$ , where $f_{i}(j):=i\cdot j\mbox{ mod }k$ , form a complete subgraph in $G_{k}$ . It is very tempting to conjecture that $p(k)$ is indeed the size of the largest clique in $G_{k}$ . We know that this is true for even $k$ and for prime $k$ . Computer search confirms that this is also true for $k=9$ .

It turns out that there is a counterexample for $k=15$ .

[TABLE]

It is found by computer search, we do not see any specific structure in it.

More intensive computer search allowed to get the examples of four functions for $k=21$ and $k=27$ . Two of four functions are again $f_{0}(x)=0$ and $f_{1}(x)=x$ , $0\leqslant x\leqslant k-1$ , (that may be always assumed a priori), and two other are, starting from the value at 0:

[TABLE]

for $k=21$ and

[TABLE]

for $k=27$ .

Denote by $\omega(k)$ the size of the largest clique in $G_{k}$ . We have the following general

Proposition 1.

[TABLE]

Proof.

Consider arbitrary corresponding functions $f_{1},\dots,f_{\omega(n)}$ , $g_{1},\dots g_{\omega(m)}$ . For $t$ from 1 to $\min(\omega(n),\omega(m))$ put

[TABLE]

Suppose that $q_{t_{1}}-q_{t_{2}}$ is not a bijection. Then for some different $a$ , $b$ we have

[TABLE]

Let $a=im+j$ , $b=rm+s$ . Then

[TABLE]

Modulo $m$ we have

[TABLE]

so $j=s$ , because $g_{t_{1}}$ and $g_{t_{2}}$ were connected in $G_{m}$ . Dividing (1) by $m$ we get

[TABLE]

Hence $i=r$ , and $a=b$ , contradiction. So $\{q_{t}\}$ is a clique of size $\min(\omega(n),\omega(m))$ . ∎

In particular, this gives a lower estimate $\omega(15^{k_{1}}21^{k_{2}}27^{k_{3}}n)\geqslant 4$ for any $n$ coprime to 6 and integers $k_{1},k_{2},k_{3}\geqslant 0$ . We expect that $\omega(3n)\geqslant 4$ for any odd $n>3$ .

Bibliography2

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] Sean Eberhard, Freddie Manners, and Rudi Mrazović. Additive triples of bijections, or the toroidal semiqueens problem. ar Xiv preprint ar Xiv:1510.05987 , 2015.
2[2] Alexander Sidorenko. On Turán problems for Cartesian products of graphs. ar Xiv preprint ar Xiv:1812.01581 , 2018.

TL;DR

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Abstract

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On a question of Sidorenko

Proposition 1**.**

Proof.

Proposition 1.