2022.06.02, Jeck Lim, Sums of linear transformations

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IBS ECOPRO Online Seminar

Jeck Lim, Sums of linear transformations
June 2 2022, Thursday @ 10:30 AM - 11:30 AM KST
Zoom

Speaker: Jeck Lim
Caltech

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.