Sums of linear transformations

We show that if \(\mathcal{L}_1\) and \(\mathcal{L}_2\) are linear transformations from \(\mathbb{Z}^d\) to \(\mathbb{Z}^d\) satisfying certain mild conditions, then, for any finite subset \(A\) of \(\mathbb{Z}^d\), $$|\mathcal{L}_1 A+\mathcal{L}_2 A|\geq \left(|\det(\mathcal{L}_1)|^{1/d}+|\det(\mat...

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Bibliographic Details
Published in:arXiv.org 2024-11
Main Authors: Conlon, David, Jeck Lim
Format: Article
Language:English
Subjects:
Linear transformations
Lower bounds
Real numbers
Online Access:Get full text
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Summary:We show that if \(\mathcal{L}_1\) and \(\mathcal{L}_2\) are linear transformations from \(\mathbb{Z}^d\) to \(\mathbb{Z}^d\) satisfying certain mild conditions, then, for any finite subset \(A\) of \(\mathbb{Z}^d\), $$|\mathcal{L}_1 A+\mathcal{L}_2 A|\geq \left(|\det(\mathcal{L}_1)|^{1/d}+|\det(\mathcal{L}_2)|^{1/d}\right)^d|A|- o(|A|).$$ 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 \(\mathcal{L}_1\) and \(\mathcal{L}_2\). As an application, we prove a lower bound for \(|A + \lambda \cdot A|\) when \(A\) is a finite set of real numbers and \(\lambda\) is an algebraic number. In particular, when \(\lambda\) is of the form \((p/q)^{1/d}\) for some \(p, q, d \in \mathbb{N}\), each taken as small as possible for such a representation, we show that $$|A + \lambda \cdot A| \geq (p^{1/d} + q^{1/d})^d |A| - o(|A|).$$ This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case \(\lambda = \sqrt{2}\).
ISSN:2331-8422