Some new results on higher energies
TL;DR: In this article, the additive energies of convex sets with small |AA| and |A(A+1) additive energies were studied and the notion of dual popular difference sets was developed.
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Abstract: In the paper we develop the method of higher energies. New upper bounds for the additive energies of convex sets, sets A with small |AA| and |A(A+1)| are obtained. We prove new structural results, including higher sumsets, and develop the notion of dual popular difference sets.
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Citations
On exponential sums over multiplicative subgroups of medium size
TL;DR: In this paper, the authors obtained new upper bounds for exponential sums over multiplicative subgroups with sizes in the range [p^c^"^" 1,p^ c^" 2, p^c" c" 2], where c"1, c"2 are some absolute constants close to 1/2.
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Energies and structure of additive sets
TL;DR: It is proved that any sumset or difference set has large E_3 energy and a full description of families of sets having critical relations between some kind of energies such as E_k, T_k and Gowers norms is given.
42
On the few products, many sums problem
TL;DR: The best known additive energy bound for real/complex sets with small multiplicative doubling is due to as discussed by the authors, which is based on combinatorial lemmata and is the best known energy bound in the prime residue field.
On higher energy decompositions and the sum-product phenomenon
George Shakan
- 01 Nov 2019
TL;DR: In this paper, the authors quantitatively improved the Balog-Wooley decomposition by partitioning A ⊂ ℝ into sets B and C such that B can be partitioned into sets C and A such that
36
New results on sum-product type growth over fields
Brendan Murphy,Giorgis Petridis,Oliver Roche-Newton,Misha Rudnev,Ilya D. Shkredov,Ilya D. Shkredov,Ilya D. Shkredov +6 more
TL;DR: In this paper, a range of new sum-product type growth estimates over a general field is presented. But the authors do not consider the problem of estimating the growth rate of the sum product.
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References
Matrix analysis: Frontmatter
Roger A. Horn,Charles R. Johnson +1 more
- 01 Jan 1985
TL;DR: This book presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications.
21.4K
A new proof of Szemerédi's theorem
TL;DR: In this paper van der Waerden showed that if the positive integers are partitioned into finitely many classes, then at least one of these classes contains arbitrarily long arithmetic progressions.
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Additive properties of product sets in fields of prime order
A.A.Glibichuk,S.V.Konyagin +1 more
TL;DR: In this paper, the field of a prime order prime order (F_p) is represented as a sum of n elements, where n is the number of elements in the field, and each element is a product of the elements from a subset of the field.
85
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Some new inequalities in additive combinatorics
TL;DR: In this paper, the additive energy of multiplicative subgroups and convex sets has been studied in the context of higher moments of convolutions, and new inequalities involving the intersections $A\cap (A-x)$ of shifts of some subset $A$ from an abelian group have been found.
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