On iterated product sets with shifts

Abstract

We prove that, for any finite set $A\subset \mathbb{Q}$ with $|AA|\le K|A|$ and any positive integer $k$, the $k$-fold product set of the shift $A+1$ satisfies the bound $$|\{(a_1+1)(a_2+1)\cdots (a_k+1):a_i\in A\}|\ge \frac{|A{|}^k}{(8k^4{)}^{kK}}.$$ 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.

References in corpus (1)

• Some remarks on sets with small quotient set

Cited by in corpus (1)

• The sum-product problem for integers with few prime factors