Results 21 to 30 of about 280 (40)

Simplices in large sets and directional expansion in ergodic actions

Forum of Mathematics, Sigma
In this paper, we study ergodic $\mathbb {Z}^r$ -actions and investigate expansion properties along cyclic subgroups. We show that under some spectral conditions, there are always directions which expand significantly a given measurable set with ...
Michael Björklund, Alexander Fish
doaj   +1 more source

On side lengths of corners in positive density subsets of the Euclidean space

, 2017
We generalize a result by Cook, Magyar, and Pramanik [3] on three-term arithmetic progressions in subsets of $\mathbb{R}^d$ to corners in subsets of $\mathbb{R}^d\times\mathbb{R}^d$.
Durcik, Polona, Kovač, Vjekoslav, Rimanić, Luka   +2 more
core   +1 more source

Asymmetric infinite sumsets in large sets of integers

Forum of Mathematics, Sigma
We show that for any set $A\subset {\mathbb N}$ with positive upper density and any $\ell ,m \in {\mathbb N}$ , there exist an infinite set $B\subset {\mathbb N}$ and some $t\in {\mathbb N}$ so that $\{mb_1 + \ell b_2 ...
Ioannis Kousek
doaj   +1 more source

Partition regularity of Pythagorean pairs

Forum of Mathematics, Pi
We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs (i.e., $x,y\in {\mathbb N}$ such that $x^2\pm y^2=z^2$ for some $z ...
Nikos Frantzikinakis, Oleksiy Klurman, Joel Moreira   +2 more
doaj   +1 more source

Boxes, extended boxes, and sets of positive upper density in the Euclidean space [PDF]

, 2020
We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: $2^n$ vertices of a fixed $n$-dimensional
Durcik, Polona, Kovač, Vjekoslav
core   +2 more sources

Finding product sets in some classes of amenable groups

Forum of Mathematics, Sigma
In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved.
Dimitrios Charamaras, Andreas Mountakis
doaj   +1 more source

Structure of multicorrelation sequences with integer part polynomial iterates along primes

, 2020
Let $T$ be a measure preserving $\mathbb{Z}^\ell$-action on the probability space $(X,{\mathcal B},\mu),$ $q_1,\dots,q_m:{\mathbb R}\to{\mathbb R}^\ell$ vector polynomials, and $f_0,\dots,f_m\in L^\infty(X)$.
Koutsogiannis, Andreas, Le, Anh N., Moreira, Joel, Richter, Florian K.   +3 more
core  

On the Davenport constant and group algebras

, 2010
For a finite abelian group $G$ and a splitting field $K$ of $G$, let $d(G, K)$ denote the largest integer $l \in \N$ for which there is a sequence $S = g_1 \cdot ... \cdot g_l$ over $G$ such that $(X^{g_1} - a_1) \cdot ... \cdot (X^{g_l} - a_l) \ne 0 \in
Smertnig, Daniel
core   +1 more source

Uniformity norms, their weaker versions, and applications

, 2022
We show that, under some mild hypotheses, the Gowers uniformity norms (both in the additive and in the hypergraph setting) are essentially equivalent to certain weaker norms which are easier to understand. We present two applications of this equivalence:
Dodos, Pandelis, Kanellopoulos, Vassilis
core  

A new proof of Sarkozy's theorem [PDF]

, 2012
It is a striking and elegant fact (proved independently by Furstenberg and Sarkozy) that in any subset of the natural numbers of positive upper density there necessarily exist two distinct elements whose difference is given by a perfect square.
Lyall, Neil
core