Related papers: On sequences without geometric progressions

This paper considers estimation of a univariate density from an individual numerical sequence. It is assumed that (i) the limiting relative frequencies of the numerical sequence are governed by an unknown density, and (ii) there is a known…

Here, we give upper and lower bounds on the count of positive integers $n\le x$ dividing the $n$th term of a nondegenerate linearly recurrent sequence with simple roots.

We prove new lower bounds on the maximum size of sets $A\subseteq \mathbb{F}_p^n$ or $A\subseteq \mathbb{Z}_m^n$ not containing three-term arithmetic progressions (consisting of three distinct points). More specifically, we prove that for…

Let the sequence S_m of nonnegative integers be generated by the following conditions: Set the first term a_0 = 0, and for all k \geq 0, let a_k+1 be the least integer greater than a_k such that no element of {a_0,...,a_k+1} is the average…

A permutation of the positive integers avoiding monotone arithmetic progressions of length $4$ with odd common difference was constructed in (LeSaulnier and Vijay, 2011). We generalise this result and show that for each $k\geq 1$, there…

Two well studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression, or in geometric progression. We study generalizations of this problem, by varying the kinds…

We describe the structure of a set of integers $A$ of positive density $\delta$, such that $A+A$ contains no squarefree integer. It turns out that the behaviour changes abruptly at the values $\delta_0=1/4-\frac{2}{\pi^2}=0.0473...$ and…

Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\leq\alpha\leq\beta\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that…

We consider real sequences $(f_n)$ that satisfy a linear recurrence with constant coefficients. We show that the density of the positivity set of such a sequence always exists. In the special case where the sequence has no positive…

In this paper we study the sets of integers which are $n$-th terms of Lucas sequences. We establish lower- and upper bounds for the size of these sets. These bounds are sharp for $n$ sufficiently large. We also develop bounds on the growth…

There exists a set $A$ of positive integers such that the number of representations of a large positive integer $m$ as a sum of two elements of $A$ grows with a lower bound of order $\log m$, but for which there is no subset $D$ of $A$…

We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic…

A permutation of the integers avoiding monotone arithmetic progressions of length $6$ was constructed in (Geneson, 2018). We improve on this by constructing a permutation of the integers avoiding monotone arithmetic progressions of length…

We prove a harmonic asymptotics density theorem for asymptotically flat initial data sets with compact boundary that satisfy the dominant energy condition. We use this to settle the spacetime positive mass theorem, with rigidity, for…

This paper is the continuation of \cite{htl}, where we deal with Lucas sequences. Here we study integers represented by integer sequences which satisfy binary recursive relations. In case of non-degenerate sequences we give bounds for the…

For an integer $b \geqslant 2$ and a set $S\subset \{0,\cdots,b-1\}$, we define the Kempner set $\mathcal{K}(S,b)$ to be the set of all non-negative integers whose base-$b$ digital expansions contain only digits from $S$. These well-studied…

It was shown by V. Bergelson that any set B with positive upper multiplicative density contains nicely intertwined arithmetic and geometric progressions: For each positive integer k there exist integers a,b,d such that $ {b(a+id)^j:i,j…

Szemer\'edi's theorem implies that there are $2^{o(n)}$ subsets of $[n]$ which do not contain a $k$-term arithmetic progression. A sparse analogue of this statement was obtained by Balogh, Morris, and Samotij, using the hypergraph container…

We prove results about uniform convergence of densities in the free central limit theorem without assumptions of boundedness on the support.

This paper is mainly concerned with sets which do not contain four-term arithmetic progressions, but are still very rich in three term arithmetic progressions, in the sense that all sufficiently large subsets contain at least one such…