Stability of sums of weighted nonnegative random variables

A stability result for sums of weighted nonnegative random variables is established and then it is utilized to obtain, among other things, a slight generalization of the Borel-Cantelli lemma and to show that the work of Jamison, Orey, and Pruitt (Z. Wahrsch. Verw. Gebiete4 (1965), 40–44) on almost sure convergence of weighted averages of independent random variables remains valid if the assumption of independence on the random variables is replaced by pairwise independence.     

On complete subsets of the cyclic group

A subset X of an abelian G is said to be complete if every element of G can be expressed as a nonempty sum of distinct elements from X.Let A⊂Zn be such that all the elements of A are coprime with n. Solving a conjecture of Erd?s and Heilbronn, Olson proved that A is complete if n is a prime and if . Recently Vu proved that there is an absolute constant c, such that for an arbitrary large n, A is complete if , and conjectured that 2 is essentially the right value of c.We show that A is complete if , thus proving the last conjecture.     

Greedy algorithm, arithmetic progressions, subset sums and divisibility

We consider a number of density problems for integer sequences with certain divisibility properties and sequences free of arithmetic progressions. Sequences of the latter type that are generated by a computer using modifications of the greedy algorithm are also provided.     

Counting subset sums of finite abelian groups

In this paper, we obtain an explicit formula for the number of zero-sum k-element subsets in any finite abelian group.     

On pseudo-random subsets of $${\mathbb Z _n}$$

The notion of pseudo-randomness of subsets of \({\mathbb Z_n}\) is defined, and the measures of pseudo-randomness are introduced. Then a construction (based on the use of hybrid character sums) will be presented for subsets of \({\mathbb Z_p}\) with strong pseudo-random properties.     

On lattice sums and Wigner limits

Wigner limits are given formally as the difference between a lattice sum, associated to a positive definite quadratic form, and a corresponding multiple integral. To define these limits, which arose in work of Wigner on the energy of static electron lattices, in a mathematically rigorous way one commonly truncates the lattice sum and the corresponding integral and takes the limit along expanding hypercubes or other regular geometric shapes. We generalize the known mathematically rigorous two- and three-dimensional results regarding Wigner limits, as laid down in [3], to integer lattices of arbitrary dimension. In doing so, we also resolve a problem posed in [6, Chapter 7]. For the sake of clarity, we begin by considering the simpler case of cubic lattice sums first, before treating the case of arbitrary quadratic forms. We also consider limits taken along expanding hyperballs with respect to general norms, and connect with classical topics such as Gauss's circle problem. Appendix A is included to recall certain properties of Epstein zeta functions that are either used in the paper or serve to provide perspective.     

关于特征和的混合均值

研究了特征和及其算术性质,并利用解析方法得到了两个关于特征和的混合均值的渐近公式,从而推广了特征和的算术性质。     

On hybrid mean value of Dedekind sums and two-term exponential sums

In this paper, we use the elementary and analytic methods to study the computational problem of one kind mean value involving the classical Dedekind sums and two-term exponential sums, and give two exact computational formulae for them.     

Multiple Dedekind sums and relative class number formulae

We construct some multiple Dedekind sums and relate them to the relative class number of an imaginary abelian number field. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On families of additive exponential sums

In this paper we compute geometric monodromy groups of additive exponential sums over BbbAⁿ. Our approach builds on work of N. Katz, and involves p-adic analysis of explicit sums and computation of the Galois group of an equation over a function field in characteristic 2. The paper also provides a brief historical outline of the problem and lists previously known results.     

On upper bounds of Chalk and Hua for exponential sums

Let be a polynomial of degree with integer coefficients, any prime, any positive integer and the exponential sum . We establish that if is nonconstant when read , then . Let , let be a zero of the congruence of multiplicity and let be the sum with restricted to values congruent to . We obtain for odd, and . If, in addition, , then we obtain the sharp upper bound .

     

On stability of sums of nonnegative random variables

We present new sufficient conditions for stability of sums of nonnegative random variables having finite moments of second order. We demonstrate that these conditions are nonimprovable in some sense. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 361, 2008, pp. 78—82.     

关于三次高斯和的均值

主要研究三次高斯和的均值性质，并给出一个较强的渐近公式。     

Classification theorems for sumsets modulo a prime

Let Zp be the finite field of prime order p and A be a subsequence of Zp. We prove several classification results about the following questions:(1) When can one represent zero as a sum of some elements of A?(2) When can one represent every element of Zp as a sum of some elements of A?(3) When can one represent every element of Zp as a sum of l elements of A?     

On Cochrane sums over short intervals

The main purpose of this paper is to study the mean square value problem of Cochrane sums over short intervals by using the properties of Gauss sums and Kloosterman sums, and finally give a sharp asymptotic formula.     

On the mean value formula of dedekind sums

The main purpose of this paper is to use the mean value theorem of the Dirichlet L-function to study the distribution property of Dedekind sums, and to give a sharper mean value formula. This work is supported by the National Natural Science Foundation of P. R. China