Estimates For Sums Research Articles

Tail Estimates for Sums of Variables Sampled by a Random Walk

We prove tail estimates for variables of the form ∑if(Xi), where (Xi)i is a sequence of states drawn from a reversible Markov chain, or, equivalently, from a random walk on an undirected graph. The estimates are in terms of the range of the function f, its variance, and the spectrum of the graph. The purpose of our estimates is to determine the number of chain/walk samples which are required for approximating the expectation of a distribution on vertices of a graph, especially an expander. The estimates must therefore provide information for fixed number of samples (as in Gillman's [4]) rather than just asymptotic information. Our proofs are more elementary than other proofs in the literature, and our results are sharper. We obtain Bernstein- and Bennett-type inequalities, as well as an inequality for sub-Gaussian variables.

On the mean value of L-functions with the weight of character sums

The main purpose of this paper is using the estimate for character sums and the analytic method to study the mean value of the Dirichlet L-functions with the weight of character sums, and give an interesting mean value theorem.

On Short Exponential Sums Involving Fourier Coefficients of Holomorphic Cusp Forms

We improve known estimates for the linear exponential sums containing Fourier coefficients of holomorphic cusp forms and show that in some cases, our bound is actually sharp. We also briefly visit nonlinear exponential sums, and prove some results concerning the density of rational numbers satisfying certain conditions.

Imaginary quadratic fields with class group exponent 5

We show that there are finitely many imaginary quadratic number fields for which the class group has exponent 5. Indeed there are finitely many with exponent at most 6. The proof is based on a method of Pierce [13]. The problem is reduced to one of counting integral points on a certain affine surface. This is tackled using the author’s “square-sieve”, in conjunction with estimates for exponential sums. The latter are derived using the q-analogue of van der Corput’s method. Mathematics Subject Classification (2000): 11R29 (11D45, 11L07, 11N36, 11LR29) Let −d be a negative fundamental discriminant and let E(−d) denote the exponent of the ideal class group of the imaginary quadratic field Q( √ −d). Under the Generalized Riemann Hypothesis one has E(−d) (log d)/(log log d), see Boyd and Kisilevsky [3], and Weinberger [15]. However without any unproved hypothesis it is not even known that E(−d) → ∞. The problem of Euler’s “Numeri Idonei” is essentially equivalent to the case E(−d) = 2, and here it has long been known that there are finitely many admissible values of d. Both Boyd and Kisilevsky loc. cit., and Weinberger loc. cit., showed that there are finitely many fundamental discriminants for which E(−d) = 3. It should be remarked that none of the proofs, either for E(−d) = 2 or for E(−d) = 3, are effective. In this paper we shall handle the case in which E(−d) = 5. Theorem 1 There is an (ineffective) constant d5 such that E(−d) 6= 5 for every fundamental discriminant −d with d > d5. While this is our principal result we also record the following easy observation. Theorem 2 Let r be a non-negative integer, and let E = 2 or E = 3.2. Then there is an (ineffective) constant dE such that E(−d) 6= E for every fundamental discriminant −d with d > dE. It follows from these two results that E(−d) ≥ 7 for all large enough d. We begin by proving Theorem 2. If E(−d) = E with E as above, then the class group takes the form C × C2r × C2a × . . .× C2b

Extension theorems for the Fourier transform associated with nondegenerate quadratic surfaces in vector spaces over finite fields

We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces on two dimensional vector spaces over finite fields. For higher dimensions, we estimate the decay of the Fourier transform of the characteristic functions on quadratic surfaces so that we obtain the Tomas–Stein exponent. Using incidence theorems, we also study the extension theorems in the restricted settings to sizes of sets in quadratic surfaces. Estimates for Gauss and Kloosterman sums and their variants play an important role.

Equations in finite fields with restricted solution sets. II (Algebraic equations)

Generalizing earlier results, it is shown that if $$ \mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D} $$ are “large” subsets of a finite field F q , then the equations a + b = cd, resp. ab + 1 = cd can be solved with $$ a \in \mathcal{A}, b \in \mathcal{B}, c \in \mathcal{C}, d \in \mathcal{D} $$ . Other algebraic equations with solutions restricted to “large” subsets of F q are also studied. The proofs are based on character sum estimates proved in Part I of the paper.

Priority sampling for estimation of arbitrary subset sums

From a high-volume stream of weighted items, we want to create a generic sample of a certain limited size that we can later use to estimate the total weight of arbitrary subsets. Applied to Internet traffic analysis, the items could be records summarizing the flows of packets streaming by a router. Subsets could be flow records from different time intervals of a worm attack whose signature is later determined. The samples taken in the past thus allow us to trace the history of the attack even though the worm was unknown at the time of sampling. Estimation from the samples must be accurate even with heavy-tailed distributions where most of the weight is concentrated on a few heavy items. We want the sample to be weight sensitive, giving priority to heavy items. At the same time, we want sampling without replacement in order to avoid selecting heavy items multiple times. To fulfill these requirements we introduce priority sampling, which is the first weight-sensitive sampling scheme without replacement that works in a streaming context and is suitable for estimating subset sums. Testing priority sampling on Internet traffic analysis, we found it to perform an order of magnitude better than previous schemes. Priority sampling is simple to define and implement: we consider a steam of items i = 0,…, n − 1 with weights w i . For each item i , we generate a random number α i ∈ (0,1] and create a priority q i = w i /α i . The sample S consists of the k highest priority items. Let τ be the ( k + 1)th highest priority. Each sampled item i in S gets a weight estimate ŵ i = max{ w i , τ}, while nonsampled items get weight estimate ŵ i = 0. Magically, it turns out that the weight estimates are unbiased, that is, E[ ŵ i ] = w i , and by linearity of expectation, we get unbiased estimators over any subset sum simply by adding the sampled weight estimates from the subset. Also, we can estimate the variance of the estimates, and find, surprisingly, that the covariance between estimates ŵ i and ŵ j of different weights is zero. Finally, we conjecture an extremely strong near-optimality; namely that for any weight sequence, there exists no specialized scheme for sampling k items with unbiased weight estimators that gets smaller variance sum than priority sampling with k + 1 items. Szegedy settled this conjecture at STOC'06.

Waring's Problem Restricted by a System of Sum of Digits Congruences

The aim of the present paper is to generalize earlier work by Thuswaldner and Tichy on Waring's problem with digital restrictions to systems of digital restrictions. Let sq (n) be the q-adic sum of digits function and let d, s, al, ml , ql ∈ N. Then for s > d2(log d + log log d + O (1)) there exists N0 ∈ N such that each integer N ≥N0 has a representation of the form N = xd 1 + ˙ ˙ ˙ + xd s where sql(xi) ≡ al mod ml (1≤ i ≤ s and 1≤ l ≤ L). The result, together with an asymptotic formula of the number of this representations, will be shown with the help of the circle method together with exponential sum estimates.

Estimation of monthly sums of total solar radiation at snow-free surface from cloud observations

An empirical formula is proposed for calculating monthly sums of total solar radiation at the snow-free surface from cloud observations and empirical parameters characterizing the optical properties of different cloud layers.

On the representation of large integers as sums of four almost equal squares of primes

In this paper, we are able to establish two localized results on the conjecture that each large integer congruent to 4 modulo 24 can be written as the sum of four squares of primes. The proof is based on the new estimates for exponential sums over primes in short intervals and a technique to get the asymptotic formula on the enlarged major arcs in the circle method.

Finite binary sequences constructed by explicit inversive methods

Some families of finite binary sequences with strong pseudo-random properties are constructed by explicit inversive methods. Two important pseudo-random measures, the well-distribution measure and the correlation measure of small order k, are evaluated for the binary sequences by using some estimates of certain exponential sums over finite fields. These constructions may provide a very attractive alternative to traditional methods in applications.

Exponential sums over definable subsets of finite fields

We prove some general estimates for exponential sums over subsets of finite fields which are definable in the language of rings. This generalizes both the classical exponential sum estimates of varieties of finite fields due to Weil, Deligne and others, and the result of Chatzidakis, van den Dries and Macintyre concerning the number of points of those definable sets. As a first application, there is no formula in the language of rings that defines for infinitely many primes an “interval” in Z/pZ that is neither bounded nor with bounded complement.

Equations in finite fields with restricted solution sets. I (Character sums)

In earlier papers, for “large” (but otherwise unspecified) subsets A, B of Zp and for h(x) ∈ Zp[x], Gyarmati studied the solvability of the equations a + b = h(x), resp. ab = h(x) with a ∈ A, b ∈ B, x ∈ Zp, and for large subsets A, B, C, D of Zp Sarkozy showed the solvability of the equations a + b = cd, resp. ab + 1 = cd with a ∈ A, b ∈ B, c ∈ C, d ∈ D. In this series of papers equations of this type will be studied in finite fields. In particular, in Part I of the series we will prove the necessary character sum estimates of independent interest some of which generalize earlier results.

A note on the upper bound estimate of high-dimensional Cochrane sums

An upper bound estimate of high-dimensional Cochrane sums is given in this note using Weinstein's version of Deligne's estimate for the hyper-Kloosterman sum and a mean value theorem of Dirichlet L-functions.

On the Partial Sums of a Fourier Series

We give sharp lower estimates for the partial sums of the Fourier series $\sin x+\frac{\cos2x}{2}+\frac{\sin 3x}{3}+\frac{\cos 4x}{4}+\cdots,$ with both an even and odd number of terms. Our results are obtained through a monotonicity property of their local minima. In particular, we prove that all partial sums are positive on $(0,\pi)$ .

Exponential sum estimates in finite commutative rings and applications

Exponential sum estimates in finite commutative rings and applications

Sum–product theorems and exponential sum bounds in residue classes for general modulus

The purpose of this Note is to extend (in the appropriate formulation) the sum–product theorem in Z q = Z / q Z (established in [J. Bourgain, N. Katz, T. Tao, A sum–product estimate in finite fields and applications, GAFA 14 (2004) 27–57; J. Bourgain, A. Glibichuk, S. Konyagin, Estimate for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. 73 (2006) 380–398] for q prime, in [J. Bourgain, M. Chang, Exponential sum estimates over subgroups and almost subgroups of Z q ∗ , where q is composite with few factors, GAFA 16 (2) (2006) 327–366] for q composite with few factors and in [J. Bourgain, A. Gamburd, P. Sarnak, Sieving and expanders, C. R. Acad. Sci. Paris, Ser. I 343 (3) (2006) 155–159] for q square free) to the case of arbitrary modulus. Consequences to exponential sum bounds (mod q) are given. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Sequences related to Legendre/Jacobi sequences

Two families of binary sequences are constructed from dth order cyclotomic generator and from dth order generalized cyclotomic generator with respect to two distinct primes respectively. By using estimates of certain exponential sums over rings or fields, the upper bounds of both the well-distribution measure and the order k (at least for small k) correlation measure of the resulting binary sequences are evaluated, and some lower bounds on the linear complexity profile of d-ary cyclotomic sequences and d-ary generalized cyclotomic sequences are presented. Our results indicate that such binary sequences possess “very good” local pseudo-randomness.

Équirépartition adélique de mesures algébriques dans un groupe résoluble et sommes de Kloosterman

This paper answers a question of Clozel and Ullmo, showing that certain sequences of adelically-defined probability measures defined on an adelic quotient of a solvable group converge to the uniform measure on that quotient. This turns out to depend on any non-trivial estimate for classical Kloosterman sums. At the end, a “horizontal” analogue of the problem is stated and solved using a result of Duke, Friedlander and Iwaniec.

Fourier Series in Control Theory-[Book review; V. Komornik and P. Loreti

This book addresses some of the following topics: observation, control, and stabilization; well-posedness in a Riesz basis setting; observation of strings; observation of beams; vector sum estimates; problems on spherical domains; multidimensional Ingham-type theorems; a general Ingham-type theorem; and problems with weakened gap condition. The book attempts to unify certain inequalities and variations offered by Ingham under the rubric of the so-called harmonic analysis method. The book gives a very clear exposition of the method of Ingham-type inequalities. Although this method is rather limited in its applicability to higher dimension problems, it does give very strong and often optimal results where it does apply.