On iterated product sets with shifts

TL;DR

This paper establishes near-optimal lower bounds for the size of the product set of a shifted finite set of rationals, using harmonic analysis tools, advancing understanding in additive combinatorics.

Contribution

It introduces a multiplicative variant of $ extLambda$-constants applied to Dirichlet polynomials to analyze shifted product sets.

Findings

Bound on the size of the $k$-fold product set of $A+1$ in terms of $|A|$, $K$, and $k$

Result is essentially optimal for certain $K$ related to $"log|A|$

New harmonic analysis method for additive combinatorics problems

Abstract

We prove that, for any finite set $A \subset \mathbb Q$ with $|AA| \leq K|A|$ and any positive integer $k$, the $k$-fold product set of the shift $A+1$ satisfies the bound $$| \{(a_1+1)(a_2+1) \cdots (a_k+1) : a_i \in A \}| \geq \frac{|A|^k}{(8k^4)^{kK}}. $$ This result is essentially optimal when $K$ is of the order $c\log|A|$, for a sufficiently small constant $c=c(k)$. Our main tool is a multiplicative variant of the $\Lambda$-constants used in harmonic analysis, applied to Dirichlet polynomials.