scholarly journals On some applications of GCD sums to Arithmetic Combinatorics

2021 ◽  
Author(s):  
Ilya D. Shkredov
Keyword(s):  
Arithmetic Combinatorics

Structure Along Arithmetic Patterns in Sequences of Vectors

2009 ◽  
Vol 18 (6) ◽  
pp. 861-870
Author(s):  
P. CANDELA
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Natural Generalization ◽  
Arithmetic Combinatorics ◽  
Vector Valued

We discuss a new direction in which the use of some methods from arithmetic combinatorics can be extended. We consider functions taking values in Euclidean space and supported on subsets of {1, 2, . . ., N}. In this context we present a proof of a natural generalization of Szemerédi's theorem. We also prove a similar generalization of a theorem of Sárkőzy using a vector-valued Fourier transform, adapting an argument of Green and obtaining effective bounds.

scholarly journals On product-one sequences over dihedral groups

2020 ◽  
pp. 2250064
Author(s):  
Alfred Geroldinger ◽  
David J. Grynkiewicz ◽  
Jun Seok Oh ◽  
Qinghai Zhong
Keyword(s):  
Finite Group ◽  
Identity Element ◽  
Finite Sequence ◽  
Finitely Generated ◽  
Dihedral Groups ◽  
Group A ◽  
Arithmetic Combinatorics ◽  
A Finite Group ◽  
Extremal Properties

Let [Formula: see text] be a finite group. A sequence over [Formula: see text] means a finite sequence of terms from [Formula: see text], where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over [Formula: see text] (with the concatenation of sequences as the operation) is a finitely generated C-monoid. Product-one sequences over dihedral groups have a variety of extremal properties. This paper provides a detailed investigation, with methods from arithmetic combinatorics, of the arithmetic of the monoid of product-one sequences over dihedral groups.

scholarly journals Klaus Friedrich Roth. 29 October 1925 — 10 November 2015

2017 ◽  
Vol 63 ◽  
pp. 487-525
Author(s):  
William Chen ◽  
Robert Vaughan
Keyword(s):  
Number Theory ◽  
Royal Society ◽  
London Mathematical Society ◽  
Imperial College ◽  
Large Sieve ◽  
Algebraic Numbers ◽  
Fields Medal ◽  
University College London ◽  
Irregularities Of Distribution ◽  
Arithmetic Combinatorics

Klaus Friedrich Roth, who died in Inverness on 10 November 2015 aged 90, made fundamental contributions to different areas of number theory, including diophantine approximation, the large sieve, irregularities of distribution and what is nowadays known as arithmetic combinatorics. He was the first British winner of the Fields Medal, awarded in 1958 for his solution in 1955 of the famous Siegel conjecture concerning approximation of algebraic numbers by rationals. He was elected a Fellow of the Royal Society in 1960, and received its Sylvester Medal in 1991. He was also awarded the De Morgan Medal of the London Mathematical Society in 1983, and elected Fellow of University College London in 1979, Honorary Fellow of Peterhouse in 1989, Honorary Fellow of the Royal Society of Edinburgh in 1993 and Fellow of Imperial College London in 1999.

scholarly journals FREIMAN THEOREM, FOURIER TRANSFORM AND ADDITIVE STRUCTURE OF MEASURES

2009 ◽  
Vol 86 (1) ◽  
pp. 97-109 ◽  
Author(s):  
A. IOSEVICH ◽  
M. RUDNEV
Keyword(s):  
Fourier Transform ◽  
Additive Structure ◽  
Arithmetic Combinatorics ◽  
Distance Problem ◽  
The Fourier Transform

AbstractWe use the Freiman theorem in arithmetic combinatorics to show that if the Fourier transform of certain measures satisfies sufficiently bad estimates, then the support of the measure possesses an additive structure. The result is then discussed in light of the Falconer distance problem.

scholarly journals A RAMSEY THEOREM ON SEMIGROUPS AND A GENERAL VAN DER CORPUT LEMMA

2016 ◽  
Vol 81 (2) ◽  
pp. 718-741 ◽  
Author(s):  
ANUSH TSERUNYAN
Keyword(s):  
Ramsey Theory ◽  
Major Theme ◽  
Multiple Recurrence ◽  
Ramsey Theorem ◽  
Arithmetic Combinatorics

AbstractA major theme in arithmetic combinatorics is proving multiple recurrence results on semigroups (such as Szemerédi’s theorem) and this can often be done using methods of ergodic Ramsey theory. What usually lies at the heart of such proofs is that, for actions of semigroups, a certain kind of one recurrence (mixing along a filter) amplifies itself to multiple recurrence. This amplification is proved using a so-called van der Corput difference lemma for a suitable filter on the semigroup. Particular instances of this lemma (for concrete filters) have been proven before (by Furstenberg, Bergelson–McCutcheon, and others), with a somewhat different proof in each case. We define a notion of differentiation for subsets of semigroups and isolate the class of filters that respect this notion. The filters in this class (call them ∂-filters) include all those for which the van der Corput lemma was known, and our main result is a van der Corput lemma for ∂-filters, which thus generalizes all its previous instances. This is done via proving a Ramsey theorem for graphs on the semigroup with edges between the semigroup elements labeled by their ratios.

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