Two random constructions inside lacunary sets

TL;DR

This paper explores the connection between the growth of integer sequences and their harmonic properties, demonstrating that certain polynomial and prime sequences contain sets with specific harmonic characteristics.

Contribution

It introduces new results on the harmonic properties of polynomial and prime sequences within lacunary sets, showing they contain sets with particular functional properties.

Findings

Polynomial sequences contain sets that are Lambda(p) for all p but not Rosenthal sets.

The sequence of primes also exhibits similar properties.

These results link sequence growth rates to harmonic and functional set properties.

Abstract

We study the relationship between the growth rate of an integer sequence and harmonic and functional properties of the corresponding sequence of characters. In particular we show that every polynomial sequence contains a set that is Lamba(p) for all p but is not a Rosenthal set. This holds also for the sequence of primes.