On Integer sequences in Product sets

TL;DR

This paper establishes bounds on the length of polynomial sequences and the number of Fibonacci numbers within product sets of finite sets of natural or complex numbers, advancing understanding of their structure.

Contribution

It provides new upper bounds for polynomial sequences in product sets and sharp bounds for Fibonacci numbers in such sets, with results applicable to both natural and complex numbers.

Findings

Bound on longest polynomial sequence in product sets

Sharp bound on Fibonacci numbers in natural number sets

Approximate bounds for Fibonacci numbers in complex number sets

Abstract

Let $B$ be a finite set of natural numbers or complex numbers. Product set corresponding to $B$ is defined by $B.B:=\{ab:a,b\in B\}$. In this paper we give an upper bound for longest length of consecutive terms of a polynomial sequence present in a product set accurate up to a positive constant. We give a sharp bound on the maximum number of Fibonacci numbers present in a product set when $B$ is a set of natural numbers and a bound which is accurate up to a positive constant when $B$ is a set of complex numbers.