On the size of $A+\lambda A$ for algebraic $\lambda$

TL;DR

This paper establishes near-optimal lower bounds for the size of sumsets involving algebraic multiples, combining additive combinatorics and geometric inequalities, and explores conjectures on the asymptotic behavior of such sumsets.

Contribution

It provides a new lower bound for sumsets with algebraic multipliers, formulates a conjecture on their asymptotic ratios, and extends results to Lebesgue measure bounds for linear transformations.

Findings

Proved a lower bound for |A+√2 A| close to optimal.

Formulated a conjecture on the limit infimum of |A+λ A|/|A| for algebraic λ.

Established a measure inequality for K+T K with applications to the conjecture.

Abstract

For a finite set $A\subset \mathbb{R}$ and real $\lambda$, let $A+\lambda A:=\{a+\lambda b :\, a,b\in A\}$. Combining a structural theorem of Freiman on sets with small doubling constants together with a discrete analogue of Pr\'ekopa--Leindler inequality we prove a lower bound $|A+\sqrt{2} A|\geq (1+\sqrt{2})^2|A|-O({|A|}^{1-\varepsilon})$ which is essentially tight. We also formulate a conjecture about the value of $\liminf |A+\lambda A|/|A|$ for an arbitrary algebraic $\lambda$. Finally, we prove a tight lower bound on the Lebesgue measure of $K+\mathcal{T} K$ for a given linear operator $\mathcal{T}\in \operatorname{End}(\mathbb{R}^d)$ and a compact set $K\subset \mathbb{R}^d$ with fixed measure. This continuous result supports the conjecture and yields an upper bound in it.