Related papers: Arithmetic-Progression-Weighted Subsequence Sums

Let $\text{AGL}(1,\Bbb F_q)$ be the affine linear group of dimension $1$ over a finite field $\Bbb F_q$. $\text{AGL}(1,\Bbb F_q)$ acts sharply 2-transitively on $\Bbb F_q$. Given $S<\text{AGL}(1,\Bbb F_q)$ and an integer $k$ with $1\le k\le…

For $(G,+)$ a finite abelian group the plus-minus weighted Davenport constant, denoted $\mathsf{D}_{\pm}(G)$, is the smallest $\ell$ such that each sequence $g_1 ... g_{\ell}$ over $G$ has a weighted zero-subsum with weights +1 and -1,…

A {\em subsequence} of a word $w$ is a word $u$ that can be obtained by deleting some letters from $w$ while maintaining the relative order of the remaining letters, e.g., $\mathtt{lala}$ is a subsequence of $\mathtt{alfalfa}$. A word, over…

We show that if a subset A of {1,...,N} does not contain any solutions to the equation x+y+z=3w with the variables not all equal, then A has size at most exp(-c(log N)^{1/7}) N, where c > 0 is some absolute constant. In view of Behrend's…

Let $\Bbbk$ be an arbitrary field and let $b > 1, n > 1$ and $a$ be three positive integers. In this paper we explicitly describe a minimal $S-$graded free resolution of the semigroup algebra $\Bbbk[S]$ when $S$ is a generalized repunit…

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}…

For a finite set $A\subset\mathbb{N}$ and $k\in \mathbb{N}$, let $\omega_k(A) = \sum_{i\in A, i\neq k}1$. For each $n\in \mathbb{N}$, define $$a_{k, n}\ =\ |\{E\subset \mathbb{N}\,:\, E = \emptyset\mbox{ or } \omega_k(E) < \min E\leqslant…

Let G be an arbitrary finite group and let S and T be two subsets such that |S|>1, |T|>1, and |TS|< |T|+|S|< |G|-1. We show that if |S|< |G|-4|G|^{1/2}+1 then either S is a geometric progression or there exists a non-trivial subgroup H such…

Let $G$ be an abelian group, and let $\mathcal F (G)$ be the free commutative monoid with basis $G$. For $\Omega \subset \mathcal F (G)$, define the universal zero-sum invariant ${\mathsf d}_{\Omega}(G)$ to be the smallest integer $\ell$…

It is proved that an irreducible quasifinite $W_\infty$-module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight $W_\infty$-module is a module of the intermediate series.…

We characterize the structure of maximum-size sum-free subsets of a random subset of an Abelian group $G$. In particular, we determine the threshold $p_c \approx \sqrt{\log n / n}$ above which, with high probability as $|G| \to \infty$,…

A subset $S$ of a group $(G,+)$ is $t$-weakly sequenceable if there is an ordering $(y_1, \ldots, y_k)$ of its elements such that the partial sums~$s_0, s_1, \ldots, s_k$, given by $s_0 = 0$ and $s_i = \sum_{j=1}^i y_j$ for $1 \leq i \leq…

A sequence in the additive group ${\mathbb Z}_n$ of integers modulo $n$ is called $n$-zero-free if it does not contain subsequences with length $n$ and sum zero. The article characterizes the $n$-zero-free sequences in ${\mathbb Z}_n$ of…

For a weight-set $A\subseteq \mathbb Z_n$, the $A$-weighted zero-sum constant $C_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in $\mathbb Z_n$ has an $A$-weighted zero-sum subsequence of…

For any finite abelian group $G$ and commutative unitary ring $R$, by $R[G]$ we denote the group algebra over $R$. Let $T=(g_1,\ldots,g_{\ell})$ be a sequence over the group $G$. We say $T$ is algebraically zero-sum free over R if…

The partial sums of a sequence ${\mathbf x} = x_1, x_2, \ldots, x_k$ of distinct non-identity elements of a group $(G,\cdot)$ are $s_0 = id_G$ and $s_j = \prod_{i=1}^j x_i$ for $0 < j \leq k$. If the partial sums are all different then…

Let A be a subset of an abelian group G. We say that A is sum-free if there do not exist x,y and z in A satisfying x + y = z. We determine, for any G, the cardinality of the largest sum-free subset of G. This equals c(G)|G| where c(G) is a…

Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of $t$-weight linear codes over ${\mathbb F}_{q}$ are presented with…

We consider certain Fibonacci-like sequences $(X_n)_{n\geq 0}$ perturbed with a random noise. Our main result is that $\frac{1}{X_n}\sum_{k=0}^{n-1}X_k$ converges in distribution, as $n$ goes to infinity, to a random variable $W$ with…

It is known that the tensor product of two sequences, in the tensor product of two separable Hilbert spaces, is a frame if and only if each component of that product is a frame. This paper proposes a sort of generalization of the…