Higher moments of convolutions

By Tomasz Schoen, Ilya D. Shkredov. 2011

92 citations recorded by the source corpus.

We study higher moments of convolutions of the characteristic function of a set, which generalize a classical notion of the additive energy. Such quantities appear in many problems of additive combinatorics as well as in number theory. In our investigation we use different approaches including basic combinatorics, Fourier analysis and eigenvalues method to establish basic properties of higher energies. We provide also a sequence of applications of higher energies additive combinatorics.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/adc442b9db21df50a6e04964a070fd18b70b4084

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In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

On the energy variant of the sum-product conjecture (2016, methodology): “f the second author and Schoen to use the third, rather the second moment, or cubic (and higher order) energy of the convolution function arising in the description of sum or product sets. See, e.g., [23]. This opportunity is inherent in the numerical values of the exponents arising in the Szemerédi”
On asymptotic formulae in some sum–product questions (2018, background): “Some further results about the properties of the energies E+k can be found in [45].”
Some new inequalities in additive combinatorics (2012, methodology): “8]. In the article we concentrate on the last method and ﬁnd new connections between the sets Ax:= A∩(A−x) and the original set A. The paper based on so–called eigenvalues method (see papers [22] and [21]) as well as Proposition 16. To obtain the proposition we develop the method from [19, 20, 24] c”
NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS (2017, methodology): “The following statement is largely part of [38, Corollary 3], see also [39].”
On sum sets of sets having small product set (2015, methodology): “In paper [7], see Lemma 27, the same result was obtained with the additional factor log(2) d(A).”