Sums of Various Dilates

Citation

Lim, Jeck (2025) Sums of Various Dilates. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6q8b-ck71. https://resolver.caltech.edu/CaltechTHESIS:05132025-055003057

Abstract

Given a finite subset A of an ambient abelian group and a dilate λ, how large must the sum of dilate A+λ∙A be in terms of A? In this thesis, we study this problem in various settings and generalizations, proving tight bounds in many cases. Our five main results are as follows.

1. In the setting of a d-dimensional subset A of ℝᵈ, we prove an exact lower bound on the size of the difference set A-A.

2. In the case when λ ∈ \in C is a transcendental number, we show that there is an absolute constant c>0 such that |A+λ∙A|≥ exp(c√log|A|)|A| for any finite subset A of C. This is best possible up to the constant c.

3. In the algebraic case, given algebraic numbers λ₁,...,λ_k, we prove tight lower bounds for the sum of dilates A+λ∙A+ ... λ_k∙A. As an important ingredient, we also prove a Freiman-type structure theorem for sets with small sums of dilates.

4. In the setting of sums of linear transformations, we prove tight bounds for the sum of two linear transformations and tight bounds for the sum of multiple pre-commuting linear transformations.

5. In the setting of groups of prime order, we prove near-optimal lower and upper bounds for the sum of dilate A+λ∙A for A of a given density and large λ.

Thesis Files