# Sums of linear transformations

### Jeck Lim (Caltech)

Thu Jun 2, 01:30-02:30 (4 weeks ago)

computer science theorycombinatorics

Audience: researchers in the topic

Comments: We show that if $L_1$ and $L_2$ are linear transformations from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ satisfying certain mild conditions, then, for any finite subset $A$ of $\mathbb{Z}^d$,

$$|L_1 A+L_2 A|\geq (|\det(L_1)|^{1/d}+|\det(L_2)|^{1/d})^d |A|- o(|A|).$$

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 many choices of $L_1$ and $L_2$. 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.

Joint work with David Conlon.

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 Organizers: O-joung Kwon*, Joonkyung Lee, Jaehoon Kim Curators: Sang-il Oum*, Hong Liu *contact for this listing

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