Related papers: Sequences of Integers with Three Missing Separatio…

We isolate conditions on the relative size of sets of natural numbers $A,B$ that guarantee a nonempty intersection $\Delta(A)\cap\Delta(B)\ne\emptyset$ of the corresponding sets of distances. Such conditions apply to a large class of zero…

We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we…

We consider avoidance of permutation patterns with designated gap sizes between pairs of consecutive letters. We call the patterns having such constraints distant patterns (DPs) and we show their relation to other pattern notions…

The independence density of a finite hypergraph is the probability that a subset of vertices, chosen uniformly at random contains no hyperedges. Independence densities can be generalized to countable hypergraphs using limits. We show that,…

We consider condensation measures of the form $P:=\frac 13 P\circ S_1^{-1}+ \frac 13 P\circ S_2^{-1}+ \frac 13 \nu $ associated with the system $(\mathcal{S}, (\frac 13, \frac 13, \frac 13), \nu) , $ where $\mathcal{S}=\{S_i\}_{i=1}^2 $ are…

Let $D$ be a digraph. A $k$-container of $D$ between $u$ and $v$, $C(u,v)$, is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u,v)$ of $D$ is a strong (resp. weak) $k^{*}$-container if there is a set of $k$…

For $\mu, \kappa$ infinite, say $\mathcal{A}\subseteq [\kappa]^\kappa$ is a $(\mu,\kappa)$-maximal independent family if whenever $\mathcal{A}_0$ and $\mathcal{A}_1$ are pairwise disjoint non-empty in $[\mathcal{A}]^{<\mu}$ then…

Suppose $k+1$ runners having nonzero constant speeds run laps on a unit-length circular track starting at the same time and place. A runner is said to be lonely if she is at distance at least $1/(k+1)$ along the track to every other runner.…

We investigate quantization coefficients for self-similar probability measures \mu on limit sets which are generated by systems S of infinitely many contractive similarities and by probabilistic vectors. The theory of quantization…

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 \colon b_1,b_2\in B\…

Let $T:[0,1]^d \rightarrow[0,1]^d$ be a piecewise expanding map with an absolutely continuous (with respect to the $d$-dimensional Lebesgue measure $m_d$) $T$-invariant probability measure $\mu$. Let $\left\{\mathbf{r}_n\right\}$ be a…

We study conditions under which integer sequences with independent, identically distributed gaps are asymptotically $k$-complete, meaning that every sufficiently large integer can be represented as the sum of exactly $k$ distinct elements…

Let $\nu$ be a vector measure defined on a $\sigma$-algebra $\Sigma$ and taking values in a Banach space. We prove that if $\nu$ is homogeneous and $L_1(\nu)$ is non-separable, then there is a vector measure $\tilde{\nu}:\Sigma \to…

Let $M$ be a positive integer and $p(n)$ be the number of partitions of a positive integer $n$. Newman's Conjecture asserts that for each integer $r$, there are infinitely many positive integers $n$ such that \[ p(n)\equiv r \pmod{M}. \]…

We study the extreme $L_p$ discrepancy of infinite sequences in the $d$-dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star $L_p$ discrepancy, which uses…

The combined universal probability M(D) of strings x in sets D is close to max_{x \in D} M({x}): their ~ logs differ by at most D's information j = I(D:H) about the halting sequence H. Thus if all x have complexity K(x) > k, D carries > i…

We investigate strong divisibility sequences and produce lower and upper bounds for the density of integers in the sequence which only have (somewhat) large prime factors. We focus on the special cases of Fibonacci numbers and elliptic…

In this paper we produce a few continuations of our previous work on partitions into fractions. Specifically, we study strictly increasing integer sequences $\{n_j\}$ such that there are partitions for all integers less than the floor of…

The combined Universal Probability M(D) of strings x in sets D is close to max M({x}) over x in D: their ~logs differ by at most D's information j=I(D:H) about the halting sequence H. Thus if all x have complexity K(x) >k, D carries >i bits…

We analyse the logical complexity and absoluteness of natural statements about Ulam sequences, with particular emphasis on the rigidity phenomena introduced by Hinman, Kuca, Schlesinger and Sheydvasser for the family $U(1,n)$. For each pair…