Related papers: Thin subbases of Piatetski-Shapiro sequences

We establish Diophantine inequalities for the fractional parts of generalized polynomials $f$, in particular for sequences $\nu(n)=\lfloor n^c\rfloor+n^k$ with $c>1$ a non-integral real number and $k\in\mathbb{N}$, as well as for $\nu(p)$…

In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+x^epsilon] is about x^epsilon/log x and the second says that the number of primes…

We show that, for any $r\geq 1$, if $g_1,\ldots,g_r$ are distinct coprime integers, sufficiently large depending only on $r$, then for any $\epsilon>0$ there are infinitely many integers $n$ such that all but $\epsilon \log n$ of the digits…

Given a (di)graph $H$, we say that a (di)graph $H^\prime$ is an $H$-subdivision if $H^\prime$ is obtained from $H$ by replacing one or more edges with internally vertex-disjoint path(s). Pavez-Sign\'{e} conjectured that for every…

We show that there is an absolute $c>0$ such that any subset of $\mathbb{F}_2^\infty$ of size $N$ is $O(N^{1-c})$-stable in the sense of Terry and Wolf. By contrast a size $N$ arithmetic progression in the integers is not $N$-stable.

Our main result shows that for every odd natural number n > 1 there is a partition of the rational numbers, x/2n in lowest terms, in the unit interval, the product of whose Gamma values takes the form of powers of 2 and pi.

We show that there are functions $f$ in the H\"older class $C^{ { \alpha }}[0,1]$, $1< { \alpha }<2$ such that $f|_{A}$ is not convex, nor concave for any $A { \subset } [0,1]$ with $ { \bar { dim }_M } A> { \alpha }-1$. Our earlier result…

We prove that a multiplicative function $f:\mathbb{N}\to\mathbb{C}$ is Toeplitz if and only if there are a Dirichlet character $\chi$ and a finite subset $F$ of prime numbers such that $f(n)=\chi(n)$ for each $n$ which is coprime to all…

We study a novel spline-like basis, which we name the "falling factorial basis", bearing many similarities to the classic truncated power basis. The advantage of the falling factorial basis is that it enables rapid, linear-time computations…

We calculate the one-level density of thin subfamilies of a family of Hecke cuspforms formed by twisting the forms in a smaller family by a character. The result gives support up to 1, conditional on GRH, and we also find several 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…

Let $(r_k)$ be an increasing sequence and $(w_k)$ a positive sequence. We study the following question: is it true that for every sequence $(a_k)$ satisfying $\sum_{k=0}^\infty |a_k|^2 w_k^2 < \infty$ there exists a function $f\in…

A set of integers is \emph{primitive} if it does not contain an element dividing another. Denote by $f(n)$ the number of maximum-size primitive subsets of $\{1,\ldots, 2n\}$. We prove that the limit $\alpha=\lim_{n\rightarrow…

A difference basis with respect to $n$ is a subset $A \subseteq \mathbb{Z}$ such that $A - A \supseteq \{1, \ldots, n\}$. R\'{e}dei and R\'{e}nyi showed that the minimum size of a difference basis with respect to $n$ is $(c+o(1))\sqrt{n}$…

Let $[\,\cdot\,]$ be the floor function. In this paper we show that whenever $\eta$ is real, the constants $\lambda_i$ satisfy some necessary conditions, then for any fixed $1<c<38/37$ there exist infinitely many prime triples $p_1,\,…

Let X be a projective curve defined over Q and t a non-constant Q-rational function on X of degree at least 2. For every integer n pick a point P_n on X such that t(P_n)=n. A result of Dvornicich and Zannier implies that, for large N, among…

In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed $1<c<\frac{427}{400}$, every sufficiently large positive number $N$ and a small constant…

An upper quasi-density on $\bf H$ (the integers or the non-negative integers) is a real-valued subadditive function $\mu^\ast$ defined on the whole power set of $\mathbf H$ such that $\mu^\ast(X) \le \mu^\ast({\bf H}) = 1$ and $\mu^\ast(k…

If $\fA$ is a unital weak-$*$ closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property $\bA_1(1)$, then the cyclic invariant subspaces index a Nevanlinna-Pick family of kernels. This yields an…

Let $f(n)$ denote the number of unordered factorizations of a positive integer $n$ into factors larger than $1$. We show that the number of distinct values of $f(n)$, less than or equal to $x$, is at most $\exp \left( C \sqrt{\frac{\log…