Sums of algebraic dilates

2025/08/26 by Conlon, David, Lim, Jeck
#05D99 #11B13 #11B75 #11P21 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.18586

Abstract

We show that if λ₁,…,λ_k are algebraic numbers, then |A+λ₁⋅ A+…+λ_k⋅ A|≥ H(λ₁,…,λ_k)|A|-o(|A|) for all finite subsets A of ℂ, where H(λ₁,…,λ_k) is an explicit constant that is best possible. The proof combines several ingredients, including a lower bound estimate on the measure of sums of linear transformations of compact sets in ℝ^d, a variant of Freiman's theorem tuned specifically to sums of dilates and the analysis of what we call lattice density, which succinctly captures how a subset of ℤ^d is arranged relative to a given flag of lattices. As an application, we revisit the study of sums of linear transformations of finite sets, in particular proving an asymptotically best possible lower bound for sums of two linear transformations.

Citations

• Brunn-Minkowski type estimates for certain discrete sumsets
• Sums of transcendental dilates
• Primes in denominators of algebraic numbers
• Sums of linear transformations in higher dimensions
• Contractions and expansion
• Sums of dilates
• John-type theorems for generalized arithmetic progressions and iterated sumsets

Related

• Sums of dilates
• Sums of dilates in ℤ_p
• Sum of dilates in vector spaces
• Sums of dilates over groups of prime order
• Sums of linear transformations