Related papers: On arithmetic progressions in self-similar sets
Unanticipated connections between different fragments of lambda calculus and different families of embedded graphs (a.k.a. "maps") motivate the problem of enumerating $\beta$-normal linear lambda terms. In this brief note, it is shown (by…
For a prime $p$, a restricted arithmetic progression in $\mathbb{F}_p^n$ is a triplet of vectors $x, x+a, x+2a$ in which the common difference $a$ is a non-zero element from $\{0,1,2\}^n$. What is the size of the largest $A\subseteq…
Let $Q^d$ be the $d$-dimensional binary hypercube. We say that $P=\{v_1,\ldots, v_k\}$ is an increasing path of length $k-1$ in $Q^d$, if for every $i\in [k-1]$ the edge $v_iv_{i+1}$ is obtained by switching some zero coordinate in $v_i$ to…
In this paper we introduce and study a certain type of sub semi-group of $\mathbb{R}/\mathbb{Z}$ which turns out to be closely related to \sz's theorem on arithmetic progressions.
An important aspect in the theory of algebras with polynomial identities is the study of the asymptotic behavior of the codimension sequence $c_n(A),\, n\geq 1,$ which measures the growth of polynomial identities of a given algebra $A$. In…
Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a random process $\{X(t), t\ge0\}$. For a given positive constant $u$, define the set of $r$th conjunctions $C_r(u):=\{t\in[0,1]: X_{r:n}(t)>u\}$ with $ X_{r:n}$ the $r$th largest…
In this paper, we construct a subset of $\mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions but has Assouad dimension 1. More precisely, we say that $F$ asymptotically and omnidirectionally contains…
We prove optimal bounds for the convergence rate of ordinal embedding (also known as non-metric multidimensional scaling) in the 1-dimensional case. The examples witnessing optimality of our bounds arise from a result in additive number…
Assume that $X$ is a connected $(q+1)$-regular undirected graph of finite order $n$. Let $A$ denote the adjacency matrix of $X$. Let $\lambda_1=q+1>\lambda_2\geq \lambda_3\geq \ldots \geq \lambda_n$ denote the eigenvalues of $A$. The…
We investigate a restriction of Paul Erdos' well-known problem from 1936 on the discrepancy of homogeneous arithmetic progressions. We restrict our attention to a finite set S of homogeneous arithmetic progressions, and ask when the…
In this paper, we consider the maximum of the $\text{Sine}_\beta$ counting process from its expectation. We show the leading order behavior is consistent with the predictions of log-correlated Gaussian fields, also consistent with work on…
In this paper, we carry out a fairly comprehensive study of two special classes of numerical semigroups, one generated by the sequence of partial sums of an arithmetic progression and the other one generated by the partial sums of a…
We present an arithmetic progression of second numbers of length 28.
We show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression A of real numbers and a sufficiently large integer n (depending on A), there needs at least Cn geometric progressions to…
In a recent result by the authors (ref. [1]) it was proved that solutions of the self-similar fragmentation equation converge to equilibrium exponentially fast. This was done by showing a spectral gap in weighted $L^2$ spaces of the…
In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers $a$ and $b$, with…
We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.
Let $M$ be a simply-connected $m$ dimensional manifold of finite type and $k$ a positif integer. In this paper we show that the rational Betti numbers of each component of the space of immersions of $M$ in $\mathbb{R}^{m+k}$, have…
We show that every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) has linear growth. This implies that the the corresponding semigroup algebra is a PI algebra.
Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several…