Related papers: Nonrepetitive sequences on arithmetic progressions

According to a classical result of Szemer\'{e}di, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson says that…

I show that a trivial modification of a standard proof of the Roth's Theorem on triples in arithmetic progression would lead to the following Theorem: If A is a "large set" that is its elements are monotone increasing integers and the sum…

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any…

A sequence $(a_1, \ldots, a_n)$ of nonnegative integers is an {\em ascent sequence} if $a_0 =0$ and for all $i \geq 2$, $a_i$ is at most 1 plus the number of ascents in $(a_1, \ldots, a_{i-1})$. Ascent sequences were introduced by…

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…

We describe new, simple, recursive methods of construction for orientable sequences over an arbitrary finite alphabet, i.e. periodic sequences in which any sub-sequence of n consecutive elements occurs at most once in a period in either…

We show that if $A=\{a_1,a_2,..., a_k\}$ is a monotone increasing set of numbers, and the differences of the consecutive elements are all distinct, then $|A+B|\geq c|A|^{1/2}|B|$ for any finite set of numbers $B$. The bound is tight up to…

A Sturmian sequence is an infinite nonperiodic string over two letters with minimal subword complexity. In two papers, the first written by Morse and Hedlund in 1940 and the second by Coven and Hedlund in 1973, a surprising correspondence…

A repetition is a sequence of symbols in which the first half is the same as the second half. An edge-coloring of a graph is repetition-free or nonrepetitive if there is no path with a color pattern that is a repetition. The minimum number…

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…

We prove game-theoretic versions of several classical results on nonrepetitive sequences, showing the existence of winning strategies using an extension of the Lov\'asz Local Lemma which can dramatically reduce the number of edges needed in…

Let $f_{s,k}(n)$ be the maximum possible number of $s$-term arithmetic progressions in a sequence $a_1<a_2<\ldots<a_n$ of $n$ integers which contains no $k$-term arithmetic progression. For all integers $k > s \geq 3$, we prove that…

A colouring of a graph is "nonrepetitive" if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive…

We study a binary Thue--Morse-type sequence arising from the base-$3/2$ expansion of integers, an archetypal automatic sequence in a rational base numeration system. Because the sequence is generated by a periodic iteration of morphisms…

A set is said to be \emph{3-free} if no three elements form an arithmetic progression. Given a 3-free set $A$ of integers $0=a_0<a_1<\cdots<a_t$, the \emph{Stanley sequence} $S(A)=\{a_n\}$ is defined using the greedy algorithm: For each…

An inversion sequence of length $n$ is an integer sequence $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck began the study of patterns in inversion sequences,…

Given a sequence of $n$ real numbers $\{S_i\}_{i\leq n}$, we consider the longest weakly increasing subsequence, namely $i_1<i_2<\dots <i_L$ with $S_{i_k} \leq S_{i_{k+1}}$ and $L$ maximal. When the elements $S_i$ are i.i.d. uniform random…

We prove a number of results related to a problem of Po-Shen Loh, which is equivalent to a problem in Ramsey theory. Let $a=(a_1,a_2,a_3)$ and $b=(b_1,b_2,b_3)$ be two triples of integers. Define $a$ to be 2-less than $b$ if $a_i<b_i$ for…

We consider exponential systems $E\left(\Lambda\right)=\left\{ e^{i\lambda t}\right\} _{\lambda\in\Lambda}$ for $\Lambda\subset\mathbb{Z}$. It has been shown by Londner and Olevskii in [9] that there exists a subset of the circle, of…

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…