Product sets cannot contain long arithmetic progressions

TL;DR

This paper establishes upper bounds on the length of arithmetic progressions within product sets of natural numbers and complex numbers, revealing limitations on their structure and providing new insights into their combinatorial properties.

Contribution

It proves new upper bounds on the longest arithmetic progression in product sets of natural and complex numbers, advancing understanding of their combinatorial structure.

Findings

Longest arithmetic progression in natural number product sets is at most O(n log^2 n / log log n)

Existence of product sets with arithmetic progressions of length Ω(n log n)

Upper bound for complex number product sets is O(n^{3/2})

Abstract

Let $B$ be a set of natural numbers of size $n$. We prove that the length of the longest arithmetic progression contained in the product set $B.B = \{bb'| \, b, b' \in B\}$ cannot be greater than $O(\frac{n\log^2 n}{\log \log n})$ and present an example of a product set containing an arithmetic progression of length $\Omega(n \log n)$. For sets of complex numbers we obtain the upper bound $O(n^{3/2})$.