Related papers: Separating Bohr denseness from measurable recurren…

We prove the existence of Rado sets in the Banach space of continuous functions on [0,1]. A countable dense set S is Rado if with probability 1, the infinite geometric random graph on S, formed by probabilistically making adjacent elements…

Assuming the well-known conjecture that [x,x+x^t] contains a prime for t > 0 and x sufficiently large, we prove: For 0 < r < 1, there exists 0 < s < r < 1, 0 < d < 1, and infinitely many primes q such that if S is a subset of Z/qZ having…

Consider the sets of integers $A$ that avoid any arrangement of $g$ congruent $h$-subsets. Our findings refine and improve upon some results by Erd\H{o}s and Harzheim about these sets.

We study dynamical notions lying between $\mathcal{U}$-frequent hypercyclicity and reiterative hypercyclicity by investigating weighted upper densities between the unweighted upper density and the upper Banach density. While chaos implies…

We study properties of a sequence $\Lambda$ obtained by a randomselection of integers $n$, where $n\in\Lambda$ with probability $\varpi_{n}$, independently of the other choices. We distinguish two cases : if…

Hereditarily indecomposable Banach spaces may have density at most continuum (Plichko-Yost, Argyros-Tolias). In this paper we show that this cannot be proved for indecomposable Banach spaces. We provide the first example of an…

In this paper we give an example of a Banach algebra $A$ and a closed ideal $I$ of $A$ such that the multiplier algebra of $I$ is equal to $A$ but $I$ does not have any bounded approximate identity. In the case that $I$ has an approximate…

We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show, a simple and natural example of a homogeneous polynomial with…

For a set $A \subset \mathbb{N}$ we characterize in terms of its density when there exists an infinite set $B \subset \mathbb{N}$ and $t \in \{0,1\}$ such that $B+B \subset A-t$, where $B+B : =\{b_1+b_2\colon b_1,b_2 \in B\}$. Specifically,…

Given a sample from some unknown continuous density $f:\mathbb{R}\to\mathbb{R}$, we construct adaptive confidence bands that are honest for all densities in a "generic" subset of the union of $t$-H\"older balls, $0<t\le r$, where $r$ is a…

We construct a set of strong recurrence which is not a van der Corput set. This shows that the class of enhanced van der Corput sets is a proper subclass of sets of strong recurrence. In addition, we derive that the class of sets of strong…

In this paper we give an example of a closed, strongly one-sided dense set which is not of uniform density type. We also show that there is a set of uniform density type which is not of strong uniform density type.

Let $ ([0,1]^d,T,\mu) $ be a measure-preserving dynamical system so that the correlations decay exponentially for H\"older continuous functions. Suppose that $ \mu $ is absolutely continuous with a density function $ h\in L^q(\mathcal L^d)…

We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) for an open set $\Omega\subset\mathbb R^n$,…

Suppose $a_n$ is a real, nonnegative sequence that does not increase exponentially. For any $p<1$ we contruct a Lebesgue measurable set $E \subseteq \mathbb{R}$ which has measure at least $p$ in any unit interval and which contains no…

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…

We randomly construct various subsets $\Lambda$ of the integers which have both smallness and largeness properties. They are small since they are very close, in various meanings, to Sidon sets: the continuous functions with spectrum in…

We answer two questions of Kra, Moreira, Richter and Robertson regarding the existence of infinite sumsets of the form $B + C$ in dense and sparse sets of integers and the relation of sumsets to sets of recurrence. We then further…

\noindent The most natural group topology on $\Z$ is the discrete one. There are other well-known group topologies on $\Z$, like the $p$-adic, defined for any prime number $p$. It is also an important group topology the weak topology with…

We observe that upper densities and spherical Federer densities may differ on all two dimensional surfaces of the sub-Riemannian Heisenberg group. This provides an entire class of intrinsic rectifiable sets having upper density strictly…