Asymptotics for the logarithm of the number of $(k,l)$-solution-free collections in an interval of naturals

A. A. Sapozhenko, V. G. Sargsyan

Lomonosov Moscow State University, 1 Leninskie Gory, 119991 Moscow, Russia

Abstract: A collection $(A_1,\dots,A_{k+l})$ of subsets of an interval $[1,n]$ of naturals is called $(k,l)$-solution-free if there is no set $(a_1,\dots,$ $a_{k+l})\in A_1\times\dots\times A_{k+l}$ that is a solution to the equation $x_1+\dots+x_k=x_{k+1}+\dots+x_{k+l}$. We obtain the asymptotics for the logarithm of the number of sets $(k,l)$-free of solutions in an interval $[1,n]$ of naturals. Bibliogr. 17.

Keywords: set, group, coset, characteristic function, progression.

UDC: 519.1

Received: 20.02.2018
Revised: 10.12.2018
Accepted: 27.02.2019

DOI: 10.33048/daio.2019.26.610

 English version: Journal of Applied and Industrial Mathematics, 2019, 13:2, 317–326

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