2022/03/18 by David Conlon, Conlon, David, Jeck Lim +1
Computer Science · Mathematics · #05D99 #11B13 #11B30 #11B75 #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2203.09827
openalex publication_date 2022/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We show that if L1 and L2 are linear transformations from ℤd to ℤd satisfying certain mild conditions, then, for any finite subset A of ℤd, |L1 A+L2 A|≥ (|det(L1)|1/d+|det(L2)|1/d)d|A|- o(|A|). This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of L1 and L2. As an application, we prove a lower bound for |A + λ⋅ A| when A is a finite set of real numbers and λ is an algebraic number. In particular, when λ is of the form (p/q)1/d for some p, q, d ∈ ℕ, each taken as small as possible for such a representation, we show that |A + λ⋅ A| ≥ (p1/d + q1/d)d |A| - o(|A|). This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case λ= √(2).