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We show that if $G$ is an amenable group and $A\subseteq G$ has positive upper Banach density, then there is an identity neighborhood $B$ in the Bohr topology on $G$ that is almost contained in $AA^{-1}$ in the sense that $B\backslash…

Erd\H{o}s showed that every set of $n$ positive integers contains a subset of size at least $n/(k+1)$ containing no solutions to $x_1 + \cdots + x_k = y$. We prove that the constant $1/(k+1)$ here is best possible by showing that if $(F_m)$…

For a fixed prime $p$, $\mathbb F_{p}$ denotes the field with $p$ elements, and $\mathbb F_{p}^{\omega}$ denotes the countable direct sum $\bigoplus_{n=1}^{\infty} \mathbb F_{p}$. Viewing $\mathbb F_{p}^{\omega}$ as a countable abelian…

Let $\mathcal{U}$ be the set of positive odd integers that cannot be represented as the sum of a prime and a power of two. In this paper, we prove that $\mathcal{U}$ is not a union of finitely many infinite arithmetic progressions and a set…

We consider the question whether there exists a Banach space $X$ of density continuum such that every Banach space of density not bigger than continuum isomorphically embeds into $X$ (called a universal Banach space of density $\cc$). It is…

We introduce the Ceiling Continued Fractions (FCT) framework for constructing three-term Egyptian fraction representations in the Erd\H{o}s-Straus conjecture. The approach exploits divisor structures of shifted integers p+i rather than…

Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $g\mapsto g+g$ has finite kernel. We establish lower bounds on the upper density of a set $A\subset G$ with respect…

The Bayesian perspective on inverse problems has attracted much mathematical attention in recent years. Particular attention has been paid to Bayesian inverse problems (BIPs) in which the parameter to be inferred lies in an…

This work is the continuation of the recent paper \cite{D2} devoted to the density-dependent incompressible Euler equations. Here we concentrate on the well-posedness issue in Besov spaces of type $B^s_{\infty,r}$ embedded in the set of…

We discuss several questions concerning sum-free sets in groups, raised by Erd\H{o}s in his survey "Extremal problems in number theory" (Proceedings of the Symp. Pure Math. VIII AMS) published in 1965. Among other things, we give a…

Assuming the generalized continuum hypothesis we construct arbitrarily big indecomposable Banach spaces. i.e., such that whenever they are decomposed as $X\oplus Y$, then one of the closed subspaces $X$ or $Y$ must be finite dimensional. It…

Let $A$ be a finite subset of an arbitrary additive group $G$, and let $\phi(A)$ denote the cardinality of the largest subset $B$ in $A$ that is sum-avoiding in $A$ (that is to say, $b_1+b_2 \not \in A$ for all distinct $b_1,b_2 \in B$).…

Let $(e_i)_i$ denote the unit vector basis of $\ell_p$, $1\leq p< \infty$, or $c_0$. We construct a reflexive Banach space with an unconditional basis that admits $(e_i)_i$ as a uniformly unique spreading model while it has no subspace with…

For a set $A$ of non-negative integers, let $R_A(n)$ denote the number of solutions to the equation $n=a+a'$ with $a$, $a'\in A$. Denote by $\chi_A(n)$ the characteristic function of $A$. Let $b_n>0$ be a sequence satisfying $\limsup_{n\to…

We construct a model of $\mathsf{MA_{\aleph_1}}+\mathsf{OCA}_T$ where Baumgartner's Axiom fails, settling a question of Farah. Moreover, in the same model there is an $\aleph_1$-dense set of reals which is neither reversible nor increasing,…

We give negative answers to two questions of Bergelson, Moreira, and Richter concerning recurrence along functions from a Hardy field. For the pair \(f_1(t)=t^{3/2}\) and \(f_2(t)=\lambda t^{3/2}+t\), where \(\lambda\in\mathbb…

The main result of the paper: Given any $\varepsilon>0$, every locally finite subset of $\ell_2$ admits a $(1+\varepsilon)$-bilipschitz embedding into an arbitrary infinite-dimensional Banach space. The result is based on two results which…

In this paper we use computational methods to disprove a conjecture by Alaoglu and Erd\H{o}s regarding the superabundant numbers.

In this paper we show that if $A$ is a subset of the primes with positive relative density $\delta$, then $A+A$ must have positive upper density $C_1\delta e^{-C_2(\log(1/\delta))^{2/3}(\log\log(1/\delta))^{1/3}}$ in $\mathbb{N}$. Our…

Let $Y$ be a compact metric space, $G$ be a group acting by transformations on $Y$. For any infinite subset $A\subset Y$, we study the density of $gA$ for $g\in G$ and quantitative density of the set $\displaystyle{\bigcup_{g\in G_n}gA}$ by…