Sums of linear transformations

Journal article

We show that if L1 and L2 are linear transformations from Zd to Zd satisfying certain mild conditions, then, for any finite subset(Equation Presented) 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 (Equation Presented) each taken as small as possible for such a representation, we show that (Equation Presented) 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.

Authors: Conlon, D; Lim, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: American Mathematical Society
• Journal: Transactions of the American Mathematical Society
• Volume: 378
• Issue: 10
• Pages: 7009-7032
• Publication date: 2025-07-31
• Acceptance date: 2024-11-18
• DOI: 10.1090/tran/9433
• EISSN: 1088-6850
• ISSN: 0002-9947
• Language: English