Let $G$ be a finite abelian group, and let $S$ be a sequence over $G$. Let
$f(S)$ denote the number of elements in $G$ which can be expressed as the sum over a nonempty subsequence of $S$. In this paper, we determine all the sequences $S$ that contains no zero-sum subsequences and $f(S)\leq 2|S|-1$.
We show that a zero-sum-free sequence of length $n$ over an abelian group spans at least $2n$ distinct subsequence sums, unless it possesses a rigid, easily-described structure.
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 $α$ ($\leq k$), let \[Σ_α (A):=\{s(B): B \subset A, |B|\geq α\}.\] Now, let $\mathcal{A}=(\underbrace{a_{1},\ldots,a_{1}}_{r_{1}~\text{copies}}, \underbrace{a_{2},\ldots,a_{2}}_{r_{2}~\text{copies}},\ldots, \underbrace{a_{k},\ldots,a_{k}}_{r_{k}~\text{copies}})$ be a finite sequence of integers with $k$ distinct terms, where $r_{i}\geq 1$ for $i=1,2,\ldots,k$. Given a subsequence $\mathcal{B}$ of $\mathcal{A}$…
We show that a finite zero-sum-free sequence $α$ over an abelian group has at least $c|α|^{4/3}$ distinct subsequence sums, unless $α$ is "controlled" by a small number of its terms; here $|α|$ denotes the number of terms of $α$, and $c>0$ is an absolute constant.
Let $\mathcal{A}$ be a sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$. For a nonnegative integer $α\leq rk$, let $Σ_α (\mathcal{A})$ be the set of all subsequence sums of $\mathcal{A}$ that correspond to the subsequences of length $α$ or more. When $r=1$, we call the subsequence sums as subset sums and we write $Σ_α (A)$ for $Σ_α (\mathcal{A})$. In this article, using some simple combinatorial arguments, we establish optimal lo…
In this paper we study the distribution of very short sequences of inversive congruential pseudorandom numbers modulo $2^t$. We derive a new bound on exponential sums with such sequences and use it to give estimate their discrepancy. The technique we use, based the method of N. M. Korobov (1972) of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford (2002), has never been used in this area and is very likely to find further applications.
Let $G$ be a finite additive abelian group with exponent $d^kn, d,n>1,$ and $k$ a positive integer. For $S$ a sequence over $G$ and $A=\{1,2,\ldots,d^kn-1\}\setminus\{d^kn/d^i:i\in[1,k]\}, $ we investigate the lower bound of the number $N_{A,0}(S)$, which denotes the number of $A$-weighted zero-sum subsequences of $S.$ In particular, we prove that $N_{A,0}(S)\ge 2^{|S|-D_A(G)+1},$ where $D_A(G)$ is the $A$-weighted Davenport Constant. We also characterize the structures of the extremal sequences for which equality holds for some groups.
Suppose $G$ is a finite abelian group and $S=g_{1}\cdots g_{l}$ is a sequence of elements in $G$. For any element $g$ of $G$ and $A\subseteq\mathbb{Z}\backslash\left\{ 0\right\} $, let $N_{A,g}(S)$ denote the number of subsequences $T=\prod_{i\in I}g_{i}$ of $S$ such that $\sum_{i\in I}a_{i}g_{i}=g$ , where $I\subseteq\left\{ 1,\ldots,l\right\}$ and $a_{i}\in A$. The purpose of this paper is to investigate the lower bound for $N_{A,0}(S)$. In particular, we prove that $N_{A,0}(S)\geq2^{|S|-D_{A}(G)+1}$, where $D_{A}(G)$ is the smallest positive integer $l$ such that every sequence over $G$ of …
Let $G$ be a finite abelian group. Let $g(G)$ be the smallest positive integer $t$ such that every subset of cardinality $t$ of the group $G$ contains a subset of cardinality $\mathrm{exp}(G)$ whose sum is zero. In this paper, we show that if X is a subset of $\mathbb{Z}^2_{2n}$ with cardinality $4n+1$ and $2n$ or $2n-1$ elements of $X$ have the same first coordintes, then $X$ contains a zero sum subset. As an application of our results we prove that $g(\mathbb{Z}^2_6) = 13.$ This settles Gao-Thangaduri's conjecture for the case $n=6.$ We also prove some results towards the general even $n$ ca…
Let $G$ be a finite additive abelian group with exponent $n$ and $S=g_{1}\cdots g_{t}$ be a sequence of elements in $G$. For any element $g$ of $G$ and $A\subseteq\{1,2,\ldots,n-1\}$, let $N_{A,g}(S)$ denote the number of subsequences $T=\prod_{i\in I}g_{i}$ of $S$ such that $\sum_{i\in I}a_{i}g_{i}=g$ , where $I\subseteq\left\{ 1,\ldots,t\right\} $ and $a_{i}\in A$. In this paper, we prove that $N_{A,0}(S)\geq2^{|S|-D_{A}(G)+1}$, when $A=\left\{ 1,\ldots,n-1\right\} $, where $D_{A}(G)$ is the smallest positive integer $l$, such that every sequence $S$ over $G$ of length at least $l$ has nonem…
Let $s_2$ be the sum-of-digits function in base $2$, which returns the number of non-zero binary digits of a nonnegative integer $n$. We study $s_2$ alon g arithmetic subsequences and show that --- up to a shift --- the set of $m$-tuples of integers that appear as an arithmetic subsequence of $s_2$ has full complexity.
We bound double sums of Kloosterman sums over a finite field ${\mathbb F}_{q}$, with one or both parameters ranging over an affine space over its prime subfield ${\mathbb F}_p \subseteq {\mathbb F}_{q} $. These are finite fields analogues of a series of recent results by various authors in finite fields and residue rings. Our results are based on recent advances in additive combinatorics in arbitrary finite field.
A formula of Kuznetsov allows one to interpret a smooth sum of Kloosterman sums as a sum over the spectrum of $GL(2)$ automorphic forms. In this paper, we construct a similar formula for the first hyper-Kloosterman sums using $GL(3)$ automorphic forms, resolving a long-standing problem of Bump, Friedberg and Goldfeld. Along the way, we develop what are apparently new bounds for the order derivatives of the classical $J$-Bessel function, and we conclude with a discussion of the original method of Bump, Friedberg and Goldfeld.
For a finite abelian group $(G,+)$, the constant $C(G)$ is defined to be the smallest natural number $k$ such that any sequence in $G$ having length $k$ will have a subsequence of consecutive terms whose sum is zero. For a subset $A\subseteq\mathbb Z_n$, the constant $C_A(n)$ is the smallest natural number $k$ such that any sequence in $G$ having length $k$ has an $A$-weighted zero-sum subsequence of consecutive terms. We determine the value of $C_A(n)$ for some particular weight-sets $A$.
We study averages over squarefree moduli of the size of exponential sums with polynomial phases. We prove upper bounds on various moments of such sums, and obtain evidence of un-correlation of exponential sums associated to different suitably unrelated and generic polynomials. The proofs combine analytic arguments with the algebraic interpretation of exponential sums and their monodromy groups.
Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly predictable despite having a large maximum order complexity. However, recent results show that polynomial subsequences of automatic sequences, such as the Thue--Morse sequence, are better candidates for pseudorandom sequences. A natural generalization of automatic sequences are morphic sequences, given by a fixed point of a prolongeable morphism that is not necess…
Sums of Kloosterman sums over primes in an arithmetic progression
对于$q$素数、$X \geq 1$和互质$u,v \in \mathbb{N}$，我们估计和\begin{equation*} \sum_{\substack{p \leq X \substack p \equiv u \hspace{-0.25cm} \mod{v} p \text{ prime}}} \text{Kl}_2(p;q), \end{equation*}，其中$\text{Kl}_2(p;q)$表示模为$q$的归一化Kloosterman和。这是一个稀疏的模拟最近的定理由于Blomer，Fouvry，Kowalski，Michel和Miliević显示取消之间的总和Kloosterman和素数在短时间间隔。我们使用的优化参数的启发Fouvry，Kowalski和米歇尔。我们的论证比较了涉及Kloosterman和的双线性型的三个不同的界。在这种方法中的第一个输入是一个双线性界，我们证明使用一致渐近的振荡积分由于Kiral，Petrow和杨。在对比的情况下，当和运行在所有素数，我们利用消除的总和固定相位积分，从Voronoi型求和的结果。第二个和第三个输入是由于Fouvry-Kowalski-Michel和Kowalski-Michel-Sawin的Kloosterman和的深双线性界。
We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$. We also conclude…
In this paper we investigate congruence relationships of particular finite generalized harmonic numbers sums. We suggest more transparent and simpler method to analyse these sums and present several additional results for certain special cases.
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of the authors. Reciprocity theorems for certain trigonometric sums are also established.
Let $G=(\mathbb Z/n\mathbb Z) \oplus (\mathbb Z/n\mathbb Z)$. Let $\mathsf {s}_{\leq k}(G)$ be the smallest integer $\ell$ such that every sequence of $\ell$ terms from $G$, with repetition allowed, has a nonempty zero-sum subsequence with length at most $k$. It is known that $\mathsf {s}_{\leq 2n-1-k}(G)=2n-1+k$ for $k\in [0,n-1]$, with the structure of extremal sequences showing this bound tight determined when $k\in \{0,1,n-1\}$, and for various special cases when $k\in [2,n-2]$. For the remaining values $k\in [2,n-2]$, the characterization of extremal sequences of length $2n-2+k$ avoiding …
