Popular Products and Continued Fractions

TL;DR

This paper establishes bounds on the popularity of products within sets exhibiting weak additive structure and applies these results to derive near-optimal bounds on the size of Zaremba's set modulo a prime.

Contribution

It introduces new bounds for product popularity in weakly additive sets and applies them to improve bounds on continued fractions, specifically Zaremba's set modulo p.

Findings

Derived bounds for product popularity in weak additive sets

Obtained nearly sharp upper bounds for Zaremba's set modulo p

Connected additive combinatorics with continued fractions

Abstract

We prove bounds for the popularity of products of sets with weak additive structure, and use these bounds to prove results about continued fractions. Namely, we obtain a nearly sharp upper bound for the cardinality of Zaremba's set modulo $p$.