КОЛЛОКВИУМ ФАКУЛЬТЕТА МАТЕМАТИКИ И КОМПЬЮТЕРНЫХ НАУК

Факультет математики и компьютерных наук, аудитория 105 (14-я линия В. О., 29)

чт. 30 января 17:15

Dmitrii Zhelezov (Alfred Renyi Institute of Mathematics, Budapest)

An analytic approach to cardinalities of sumsets

The aim of this study is to understand the nature of structures in Z^{d}, the presence of which implies that the sumset must be large. The archetype is Freiman’s theorem that if a set A ⊂ Z d is proper d-dimensional, then

|A+A|\geq (d+1)|A|-\binom{d+1}{2}.

The assumption on dimension can be expressed as S_{d} ⊂ A for a d-dimensional simplex S_{d}. In general, the induced doubling of a set U is the quantity

\inf\limits_{A\supset U}\frac{|A+A|}{|A|};

our main aim is to give lower estimates for it and related quantities. Applications for the sum-product problem, related to the work of [BC04], will be the subject of another paper. While our main interest is in Z^{d}, we shall mostly formulate our results for general, typically torsion-free commutative groups. Since we work with finite sets and a finitely generated torsion-free group is isomorphic to some Z^{d}, it is not more general, but we rarely need the coordinates. In the first part we work with sets, in the second part we study a weighted version which will be necessary for the proof of the main results. By introducing a weighted analog, we will be able to use tensorization: that is we prove a d-dimensional inequality by induction on dimension alongside a two point inequality. This is a method commonly used in analysis, for instance in the Pr´ekopa-Leindler inequality [Pr´e71] and Beckner’s inequality [Bec75]. We discuss this more below, but also invite the reader to the excellent survey paper of Gardner [Gar02].

[BC04] Jean Bourgain and Mei-Chu Chang. On the size of k-fold sum and product sets of integers. Journal of the American Mathematical Society, 17(2):473–497, 2004.

[Bec75] William Beckner. Inequalities in Fourier analysis. PhD thesis, Princeton., 1975. [Gar02] Richard Gardner. The Brunn–Minkowski inequality. Bulletin of the American Mathematical Society, 39(3):355–405, 2002.

[Gar02] Richard Gardner. The Brunn–Minkowski inequality. Bulletin of the American Mathematical Society, 39(3):355–405, 2002.

[Pr´e71] Andr´as Pr´ekopa. Logarithmic concave measures with application to stochastic programming. Acta Scientiarum Mathematicarum, 32:301–316, 1971.