Related papers: Structures in Additive Sequences

We identify a binary sequence $\mathcal{S}=(s_n)_{n=0}^\infty$ with the $2$-adic integer $G_\mathcal{S}(2)=\sum\limits_{n=0}^\infty s_n2^n$. In the case that $G_\mathcal{S}(2)$ is algebraic over $\mathbb{Q}$ of degree $d\ge 2$, we prove…

We extend our investigation of $2$-determinants, which we defined in a previous paper. For a linear homogenous recurrence of the second order, we consider relations between different sequences satisfying the same linear homogeneous…

Zeckendorf's Theorem states that every positive integer can be uniquely represented as a sum of non-adjacent Fibonacci numbers, indexed from $1, 2, 3, 5,\ldots$. This has been generalized by many authors, in particular to constant…

A classical argument was introduced by Khintchine in 1926 in order to exhibit the existence of totally irrational singular linear forms in two variables. This argument was subsequently revisited and extended by many authors. For instance,…

A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are either 0, 1 or -1. The only known Barker sequences have length 2, 3, 4, 5, 7, 11 or 13. It is an old conjecture that no longer Barker sequences…

Let V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-coloring of {1,2,...,n}. We show that (1675/32768) n^2 (1+o(1)) <= V(n) <= (117/2192) n^2(1+o(1)). As a consequence, we find that V(n) is strictly…

We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that…

We give a vector identity for $n+2$ points in $\mathbb R^n$. It follows as a corollary that when $n$ is odd the sum of the signed volumes of the $n$-simplices is zero, and when $n$ is even, the alternating sum of the signed volumes is zero.

We construct large subsets of the first $N$ positive integers which avoid certain arithmetic configurations. In particular, we construct a set of order $N^{0.7685}$ lacking the configuration $\{x,x+y,x+y^2\},$ surpassing the $N^{3/4}$ limit…

In this paper we prove that every set $A\subset\mathbb{Z}$ satisfying the inequality $\sum_{x}\min(1_A*1_A(x),t)\le(2+\delta)t|A|$ for $t$ and $\delta$ in suitable ranges, then $A$ must be very close to an arithmetic progression. We use…

In this paper we continue to investigate the properties of those sequences $\{a_n\}$ satisfying the condition $\sum_{k=0}^n\binom nk(-1)^ka_k=\pm a_n$ $(n\ge 0)$. As applications we deduce new recurrence relations and congruences for…

A permutation is so-called two stack sortable if it (i) avoids the (scattered) pattern 2-3-4-1, and (ii) contains a 3-2-4-1 pattern only as part of a 3-5-2-4-1 pattern. Here we show that the permutations on [n] satisfying condition (ii)…

It is known that, when $n$ is even, the number of permutations of $\{1,2,\dots,n\}$ all of whose cycles have odd length equals the number of those all of whose cycles have even length. Adin, Heged\H{u}s and Roichman recently found a…

We prove that every set $A\subset\mathbb{Z}/p\mathbb{Z}$ with $\mathbb{E}_x\min(1_A*1_A(x),t)\le(2+\delta)t\mathbb{E}_x 1_A(a)$ is very close to an arithmetic progression. Here $p$ stands for a large prime and $\delta,t$ are small real…

In this paper we investigate some new problems in additive combinatorics. Our problems mainly involve permutations (or circular permutations) $n$ distinct numbers (or elements of an additive abelian group) $a_1,\ldots,a_n$ with adjacent…

Let $p$ be any odd prime number. Let $k$ be any positive integer such that $2\leq k\leq [\frac{p+1}3]+1$. Let $S = (a_1,a_2,...,a_{2p-k})$ be any sequence in ${\Bbb Z}_p$ such that there is no subsequence of length $p$ of $S$ whose sum is…

We show that if $A$ is a subset of a group of prime order $p$ such that $|2A|<2.7652|A|$ and $|A|<1.25\cdot10^{-6}p$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms, and $2A$ contains an arithmetic…

Experimental results show that, when the order $n$ is odd, there are de Bruijn sequences such that the corresponding complement sequence and the reverse sequence are the same. In this paper, we propose one efficient method to generate such…

Fibonacci sequence, generated by summing the preceding two terms, is a classical sequence renowned for its elegant properties. In this paper, leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of…

We prove that every sufficiently large odd integer is a sum of two positive squares and a prime. Let R(n) be the number of representations n = x^2 + y^2 + p with x, y >= 1 and p prime. We show that R(n) > 0 for all odd n >= n0 and obtain…