Related papers: On distinct consecutive differences

We study the number of $s$-element subsets $J$ of a given abelian group $G$, such that $|J+J|\leq K|J|$. Proving a conjecture of Alon, Balogh, Morris and Samotij, and improving a result of Green and Morris, who proved the conjecture for $K$…

Let $S$ be the numerical semigroup generated by three consecutive numbers $a,a+1,a+2$, where $a\in\mathbb{N}$, $a\geq 3$. We describe the elements of $S$ whose factorizations have all the same length, as well as the set of factorizations of…

In this paper upper bounds are given for the successive differences $A_{n+1}-A_{n}$ and B$_{n}-B_{n-1}$ where $A_{n}=1/(n-1) \tsum_{r=1}^{n-1}f(r/n)$, $B_{n}=1/(n+1) \tsum_{r=0}^{n}f(r/n)$ and $f$ is superquadratic function. We obtain…

We show that if A is a finite set of integers then it has a subset S of size \log^{1+c} |A| (c>0 absolute) such that s+s' is never in A when s and s' are distinct elements of S.

The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…

An $(a,b)$-difference necklace of length $n$ is a circular arrangement of the integers $0, 1, 2, \ldots , n-1$ such that any two neighbours have absolute difference $a$ or $b$. We prove that, subject to certain conditions on $a$ and $b$,…

We construct the base $2$ expansion of an absolutely normal real number $x$ so that, for every integer $b$ greater than or equal to $2$, the discrepancy modulo $1$ of the sequence $(b^0 x, b^1 x, b^2 x , \ldots)$ is essentially the same as…

We consider some coloring issues related to the famous Erd\H {o}s Discrepancy Problem. A set of the form $A_{s,k}=\{s,2s,\dots,ks\}$, with $s,k\in \mathbb{N}$, is called a \emph{homogeneous arithmetic progression}. We prove that for every…

We prove decidability results on the existence of constant subsequences of uniformly recurrent morphic sequences along arithmetic progressions. We use spectral properties of the subshifts they generate to give a first algorithm deciding…

Let $A = \{0 = a_0 < a_1 < \cdots < a_{\ell + 1} = b\}$ be a finite set of non-negative integers. We prove that the sumset $NA$ has a certain easily-described structure, provided that $N \geqslant b-\ell$, as recently conjectured by Shakan…

Let~$A$ be a set of nonnegative integers. Let~$(h A)^{(t)}$ be the set of all integers in the sumset~$hA$ that have at least~$t$ representations as a sum of~$h$ elements of~$A$. In this paper, we prove that, if~$k \geq 2$,…

We consider the set $$\mathcal{A} = \left\{10\cdot a + 11\cdot b \ | \gcd(a,b)=1, a\geq 1, b\geq 2a+1 \right\}.$$ We will prove that $\mathcal{A}$ is unbounded and that there exists a natural number $M\notin \mathcal{A}$ for which…

The main result of the paper is the following. Let a non-degenerate distribution have finite moments $\mu_k$ of all orders $k=0,1,2,\ldots$. Then the sequence $\{\mu_k/k!, \; k=0,1,2,\ldots\}$ either contains infinitely many different terms…

Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha}…

In a monotonic sequence game, two players alternately choose elements of a sequence from some fixed ordered set. The game ends when the resulting sequence contains either an ascending subsequence of length a or a descending one of length d.…

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…

The lambda-dilate of a set A is lambda*A={lambda a : a \in A}. We give an asymptotically sharp lower bound on the size of sumsets of the form lambda_1*A+...+lambda_k*A for arbitrary integers lambda_1,...,lambda_k and integer sets A. We also…

In the paper, we search for monochromatic infinite additive structures involving polynomials over $\mathbb{N}$. It is proved that for any $r\in \mathbb{N}$, any two distinct natural numbers $a,b$, and any $2$-coloring of $\mathbb{N}$, there…

The investigation of primes in certain arithmetic sequences is one of the fundamental problems in number theory and especially, finding blocks of distinct primes has gained a lot of attention in recent years. In this context, we prove the…

For any set $A$ of natural numbers with positive upper Banach density, we show the existence of an infinite set $B$ and sequences $(t_k)_{k\in \mathbb{N}}, (s_k)_{k\in \mathbb{N}}$ of natural numbers such that $\left\{ \sum_{n \in F}n : F…