Sums of algebraic dilates

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Given a real number λ and a finite set A of real numbers, how small can the size of the sum of dilate A + λ.A be in terms of |A|? If λ is transcendental, then |A + λ.A| grows superlinearly in |A|, whereas if λ is algebraic, then |A + λ.A| only grows linearly in |A|. There have been several works in recent years to prove optimal linear bounds in the algebraic case, but tight bounds were only known when λ is an algebraic integer or of the form (p/q)^{1/d}.

In this talk, we prove tight bounds for sums of arbitrarily many algebraic dilates |A + λ1.A + ... + λk.A|. We will discuss the main tools used in the proof, which include a Frieman-type structure theorem for sets with small sums of dilates, and a high-dimensional notion of density which we call "lattice density". Joint work with David Conlon.

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