Results 11 to 20 of about 93 (44)
Sumsets and entropy revisited [PDF]
Random Structures &Algorithms, Volume 66, Issue 1, January 2025.The entropic doubling σ ent [ X ] $$ {\sigma}_{\mathrm{ent}}\left[X\right] $$ of a random variable X $$ X $$ taking values in an abelian group G $$ G $$ is a variant of the notion of the doubling constant σ [ A ] $$ \sigma \left[A\right] $$ of a finite ...
Green, Ben +2 more
core +3 more sources
A strengthening of Freiman's 3k−4$3k-4$ theorem
Bulletin of the London Mathematical Society, Volume 55, Issue 5, Page 2363-2381, October 2023., 2023 Abstract In its usual form, Freiman's 3k−4$3k-4$ theorem states that if A$A$ and B$B$ are subsets of Z${\mathbb {Z}}$ of size k$k$ with small sumset (of size close to 2k$2k$), then they are very close to arithmetic progressions. Our aim in this paper is to strengthen this by allowing only a bounded number of possible summands from one of the sets.
Béla Bollobás +2 more
wiley +1 more source
Diameter‐free estimates for the quadratic Vinogradov mean value theorem
Proceedings of the London Mathematical Society, Volume 126, Issue 1, Page 76-128, January 2023., 2023 Abstract Let s⩾3$s \geqslant 3$ be a natural number, let ψ(x)$\psi (x)$ be a polynomial with real coefficients and degree d⩾2$d \geqslant 2$, and let A$A$ be some large, non‐empty, finite subset of real numbers. We use Es,2(A)$E_{s,2}(A)$ to denote the number of solutions to the system of equations ∑i=1s(ψ(xi)−ψ(xi+s))=∑i=1s(xi−xi+s)=0,$$\begin ...
Akshat Mudgal
wiley +1 more source
Journal of the London Mathematical Society, Volume 106, Issue 4, Page 2927-2958, December 2022., 2022 Abstract In this paper, we study additive properties of finite sets of lattice points on spheres in three and four dimensions. Thus, given d,m∈N$d,m \in \mathbb {N}$, let A$A$ be a set of lattice points (x1,⋯,xd)∈Zd$(x_1, \dots , x_d) \in \mathbb {Z}^d$ satisfying x12+⋯+xd2=m$x_1^2 + \dots + x_{d}^2 = m$.
Akshat Mudgal
wiley +1 more source
Distributions and wave front sets in the uniform non‐archimedean setting
Transactions of the London Mathematical Society, Volume 5, Issue 1, Page 97-131, December 2018., 2018 Abstract We study some constructions on distributions in a uniform p‐adic context, and also in large positive characteristic, using model theoretic methods. We introduce a class of distributions which we call distributions of C exp ‐class and which is based on the notion of C exp ‐class functions from Cluckers and Halupczok [J. Ecole Polytechnique (JEP)
Raf Cluckers +3 more
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Analytic Erdös‐Turán conjectures and Erdös‐Fuchs theorem
International Journal of Mathematics and Mathematical Sciences, Volume 2005, Issue 23, Page 3767-3780, 2005., 2005 We consider and study formal power series, that we call supported series, with real coefficients which are either zero or bounded below by some positive constant. The sequences of such coefficients have a lot of similarity with sequences of natural numbers considered in additive number theory.
L. Haddad, C. Helou, J. Pihko
wiley +1 more source
The structure of sets with cube‐avoiding sumsets
Mathematika, Volume 71, Issue 4, October 2025.Abstract Suppose G$G$ is a finite abelian group, Z0⊂G$Z_0 \subset G$ is not contained in any strict coset in G$G$, and E,F$E,F$ are dense subsets of Gn$G^n$ such that the sumset E+F$E+F$ avoids Z0n$Z_0^n$. We show that E$E$ and F$F$ are almost entirely contained in sets defined by a bounded number of coordinates, that is, sets E′×GIc$E^{\prime } \times
Thomas Karam, Peter Keevash
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Strong External Difference Families and Classification of α‐Valuations
Journal of Combinatorial Designs, Volume 33, Issue 9, Page 343-356, September 2025.ABSTRACT One method of constructing ( a 2 + 1 , 2 , a , 1 )‐SEDFs (i.e., strong external difference families) in Z a 2 + 1 makes use of α‐valuations of complete bipartite graphs K a , a. We explore this approach and we provide a classification theorem which shows that all such α‐valuations can be constructed recursively via a sequence of “blow‐up ...
Donald L. Kreher +2 more
wiley +1 more source
Infinite unrestricted sumsets of the form B+B$B+B$ in sets with large density
Bulletin of the London Mathematical Society, Volume 57, Issue 1, Page 48-68, January 2025.Abstract For a set A⊂N$A \subset {\mathbb {N}}$, we characterize the existence of an infinite set B⊂N$B \subset {\mathbb {N}}$ and t∈{0,1}$t \in \lbrace 0,1\rbrace$ such that B+B⊂A−t$B+B \subset A-t$, where B+B={b1+b2:b1,b2∈B}$B+B =\lbrace b_1+b_2\colon b_1,b_2 \in B\rbrace$, in terms of the density of the set A$A$. Specifically, when the lower density
Ioannis Kousek, Tristán Radić
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The structure and density of k$k$‐product‐free sets in the free semigroup and group
Journal of the London Mathematical Society, Volume 111, Issue 1, January 2025.Abstract The free semigroup F$\mathcal {F}$ on a finite alphabet A$\mathcal {A}$ is the set of all finite words with letters from A$\mathcal {A}$ equipped with the operation of concatenation. A subset S$S$ of F$\mathcal {F}$ is k$k$‐product‐free if no element of S$S$ can be obtained by concatenating k$k$ words from S$S$, and strongly k$k$‐product‐free ...
Freddie Illingworth +2 more
wiley +1 more source