Related papers: Universally L^1-Bad Arithmetic Sequences

We answer a question of J. Bourgain. We show that the sequence n^2 is L^1-universally bad.

A universally L^1 good sequence n_k is constructed with n_{k+1}-n_k tending to infinity. For ergodic transformations non-conventional ergodic averages of L^1 functions computed by using this sequence converge to the integral of the…

We study convergence of ergodic averages along squares with polynomial weights. For a given polynomial $P\in \mathbb{Z}[\cdot]$, consider the set of all $\theta\in[0,1)$ such that for every aperiodic system $(X,\mu, T)$ there is a function…

Examples are constructed of sparse subsequences of the integers for which the associated maximal averages operator is of weak type (1,1). A consequence, by transference, is that an almost everywhere L^1 -- type ergodic theorem holds for…

We consider a generalization of Euclid's proof of the infinitude of primes and show that it leads to variants of the Euclid-Mullin sequence that provably contain every prime number.

We prove an L^1 subsequence ergodic theorem for sequences chosen by independent random selector variables, thereby showing the existence of universally L^1-good sequences nearly as sparse as the set of squares. In the process, we prove that…

In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic…

We give a new characterization of generalized B\"uchi sequences (sequences whose sequence of squares has constant second difference $(a)$, for some fixed integer $a$) of length 3 over the integers and a strategy for attacking B\"uchi's n…

We introduce a new class of pseudoprimes-so called "overpseudoprimes" which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n, we show that odd number n is overpseudoprime iff value…

We prove under the Bombieri-Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequences which are not formed by the squares of integers in…

In this short paper, we prove, by only using elementary tools, general cases when $U_n(P,Q) \neq \square$, where $U_n(P,Q)$ is the Lucas sequence of the first type.

Suppose that $\alpha,\beta\in\mathbb{R}$. Let $\alpha\geqslant1$ and $c$ be a real number in the range $1<c< 12/11$. In this paper, it is proved that there exist infinitely many primes in the generalized Piatetski--Shapiro sequence, which…

Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q), n=0,1,2,... is defined by U_0=0, U_1=1, U_n= P U_{n-1}-Q U_{n-2} for n>1. For each positive integer n<8 we describe all Lucas sequences with (P,Q)=1 having the property that…

We study whether several consecutive prime gaps can all be relatively large at the same time, or is it possible that all are squares or perfect powers, or perhaps none of them are squares? A few related results and problems are also…

Let $ \prod_{i=1}^d (X-\alpha_i Y) \in{\mathbb C}[X,Y]$ be a binary form and let $\epsilon_1,\dots,\epsilon_d$ be nonzero complex numbers. We consider the family of binary forms $ \prod_{i=1}^d (X-\alpha_i \epsilon_i^aY)$, $a\in {\mathbb…

Motivated by questions of Fouvry and Rudnick on the distribution of Gaussian primes, we develop a very general setting in which one can study inequities in the distribution of analogues of primes through analytic properties of infinitely…

We establish a strong form of Littlewood's conjecture with inhomogeneous shifts, for a full-dimensional set of pairs of badly approximable numbers on a vertical line. We also prove a uniform assertion of this nature, generalising a strong…

Let $(U_n)_{n\geq 0}$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $\ell$, let $\ell \cdot 2^{\ell} + 1$ be a Cullen number. Recently in \cite{bmt}, generalized Cullen numbers in…

Generalized L\"uroth series generalize $b$-adic representations as well as L\"uroth series. Almost all real numbers are normal, but it is not easy to construct one. In this paper, a new construction of normal numbers with respect to…

We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences like the sum of digits in base $q$ modulo $m$, the Rudin-Shapiro sequence and some generalizations. This gives, for any base, a class…