Related papers: Erd\H{o}s similarity problem via bi-Lipschitz embe…
New partial results are obtained related to the following old problem of Erd\"os: for any infinite set $X$ of real numbers to show that there is always a measurable (or, equivalently, closed) subset of reals of positive Lebesgue measure…
A conjecture of Erd\H{o}s states that for any infinite set $A \subseteq \mathbb R$, there exists $E \subseteq \mathbb R$ of positive Lebesgue measure that does not contain any nontrivial affine copy of $A$. The conjecture remains open for…
We study some variants of the Erd\H{o}s similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset…
We show that for any infinite set $A$ in ${\mathbb R}$, there exists a compact set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that does not contain any non-trivial affine copy of $A$. This proves the Erd\"os similarity…
In relation to the Erd\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are…
Erd\H{o}s conjectured that for any set $A\subseteq \mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,C\subseteq \mathbb{N}$ such that $B+C\subseteq A$. We verify Erd\H{o}s' conjecture in the case that $A$ has…
Erd\H{o}s posed the question whether there exist infinitely many sets of consecutive numbers whose least common multiple (lcm) exceeds the lcm of another, larger set with greater consecutive numbers. In this paper, we answer this question…
Erd\H{o}s similarity conjecture was proposed by P. Erd\H{o}s in 1974. The conjecture remains open for exponentially decaying sequences as well as Cantor sets that have both Newhouse thickness and Hausdorff dimension zero. In this article,…
The Erd\H{o}s discrepancy problem, now a theorem by T. Tao, asks whether every sequence with values plus or minus one has unbounded discrepancy along all homogeneous arithmetic progressions. We establish weighted variants of this problem,…
In a recent paper, Kolountzakis and Papageorgiou ask if for every $\epsilon \in (0,1]$, there exists a set $S \subseteq \mathbb{R}$ such that $\vert S \cap I\vert \geq 1 - \epsilon$ for every interval $I \subset \mathbb{R}$ with unit…
We show that there exists a strong uniform embedding from any proper metric space into any Banach space without cotype. Then we prove a result concerning the Lipschitz embedding of locally finite subsets of $\mathcal{L}_{p}$-spaces. We use…
We develop a new multi-scale framework flexible enough to solve a number of problems involving embedding random sequences into random sequences. Grimmett, Liggett and Richthammer asked whether there exists an increasing M-Lipschitz…
It is known by a result of Mendes and Sampaio that the Lipschitz normal embedding of a subanalytic germ is fully characterized by the Lipschitz normal embedding of its link. In this note, we show that the result still holds for definable…
Metric embedding is a powerful tool used extensively in mathematics and computer science. We devise a new method of using metric embeddings recursively, which turns out to be particularly effective in $\ell_p$ spaces, $p>2$, yielding…
It is known that if finite subsets of a locally finite metric space $M$ admit $C$-bilipschitz embeddings into $\ell_p$ $(1\le p\le \infty)$, then for every $\epsilon>0$, the space $M$ admits a $(C+\epsilon)$-bilipschitz embedding into…
This paper is concerned with embeddings of homogeneous spaces into Euclidean spaces. We show that any homogeneous metric space can be embedded into a Hilbert space using an almost bi-Lipschitz mapping (bi-Lipschitz to within logarithmic…
Let X and Y be two infinite-dimensional Banach spaces. If X is crudely finitely representable in every finite-codimensional subspace of Y, then any proper subset of X almost bi-Lipschitz embeds into Y, in a sense quite close to that of F.…
We give sufficient conditions for a metric space to bilipschitz embed in L_1. In particular, if X is a length space and there is a Lipschitz map u:X--->R such that for every interval I in R, the connected components of the inverse image…
The present paper is devoted to the study of the well-posedness issue for the density-dependent Euler equations in the whole space. We establish local-in-time results for the Cauchy problem pertaining to data in the Besov spaces embedded in…
In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erd\H{o}s. The proof features two different decompositions of…