Estimates For Exponential Sums Research Articles

From oscillatory integrals to complete exponential sums

In (8), Phong and Stein establish a sharp and stable bound for (one dimensional) scalar oscillatory integrals with a polynomial phasein terms of root clusters of the derivative � ' . In this note we prove an analogous result for complete exponential sums. When one considers only singleton clusters, the corresponding estimate for exponential sums was established by Loxton and Vaughan in (5). Considering all possible clusters containing a particular root allows one to obtain bounds for exponential sums which are stable under perturbations of the phase.

Bounds for the cubic Weyl sum

Subject to the abc-conjecture, we improve the standard Weyl estimate for cubic exponential sums in which the argument is a quadratic irrational. Specifically. we show that $$ \sum\limits_{n \leqslant N} {e\left( {\alpha {n^3}} \right){ \ll_{\varepsilon, \alpha }}{N^{\tfrac{5}{7} + \varepsilon }}} $$ for any $$ \varepsilon > 0 $$ and any quadratic irrational $$ \alpha \in \mathbb{R} - \mathbb{Q} $$ . Classically one would have had the (unconditional) exponent $$ \frac{3}{4} + \varepsilon $$ for such α. Bibliography: 5 titles.

Enlarged major arcs in additive problems

Enlarged major arcs in the Waring-Goldbach problem are studied by using large sieve estimates for Dirichlet polynomials and estimates for exponential sums over primes.

Small solutions of polynomial congruences

Let p be prime and q|p − 1. Suppose x q ≡ a(mod p) has a solution. We estimate the size of the smallest solution x 0 with 0 < x 0 < p. We prove that |x 0| ≪ p 3/2 q −1 log p. By applying the Burgess character sum estimates, and estimates of certain exponential sums due to Bourgain, Glibichuk and Konyagin, we derive refinements of our result.

Pseudorandom numbers and hash functions from iterations of multivariate polynomials

Dynamical systems generated by iterations of multivariate polynomials with slow degree growth have proved to admit good estimates of exponential sums along their orbits which in turn lead to rather stronger bounds on the discrepancy for pseudorandom vectors generated by these iterations. Here we add new arguments to our original approach and also extend some of our recent constructions and results to more general orbits of polynomial iterations which may involve distinct polynomials as well. Using this construction we design a new class of hash functions from iterations of polynomials and use our estimates to motivate their “mixing” properties.

Large Family of Sequences from Elliptic Curves over Residue Class Rings

An upper bound is established for certain exponential sums on the rational points of an elliptic curve over a residue class ring $\mathbb{Z}_N$, N=pq for two distinct odd primes p and q. The result is a generalization of an estimate of exponential sums on rational point groups of elliptic curves over finite fields. The bound is applied to showing the pseudorandomness of a large family of binary sequences constructed by using elliptic curves over $\mathbb{Z}_N$.

On certain exponential sums related to [formula omitted] cusp forms

We will prove a sharp estimate for certain exponential sums of Fourier coefficients of GL ( 3 ) cusp forms.

Exponential sums: Questions by Denef, Sperber, and Igusa

We prove the remaining part of the conjecture by Denef and Sperber [Denef, J. and Sperber, S., Exponential sums mod p n and Newton poly-hedra, Bull. Belg. Math. Soc., suppl. (2001) 55-63] on nondegenerate local exponential sums modulo p m . We generalize Igusa's conjecture in the introduction of [Igusa, J., Lectures on forms of higher degree, Lect. Math. Phys., Springer-Verlag, 59 (1978)] from the homogeneous to the quasi-homogeneous case and prove the nondegenerate case as well as the modulo p case. We generalize some results by Katz in [Katz, N. M., Estimates for singular exponential sums, Internat. Math. Res. Notices (1999) no. 16, 875-899] on finite field exponential sums to the quasi-homogeneous case.

Local Zeta Functions Supported on Analytic Submanifolds and Newton Polyhedra

The local zeta functions (also called Igusa's zeta functions) over p-adic fields are connected with the number of solutions of congruences and exponential sums mod pm. These zeta functions are defined as integrals over open and compact subsets with respect to the Haar measure. In this article, we introduce new integrals defined over submanifolds, or more generally, over nondegenerate complete intersection varieties, and study their connections with some arithmetical problems such as estimation of exponential sums mod pm. In particular, we extend Igusa's method for estimating exponential sums mod pm to the case of exponential sums mod pm along nondegenerate smooth varieties.

The Erdös–Falconer Distance Problem, Exponential Sums, and Fourier Analytic Approach to Incidence Theorems in Vector Spaces over Finite Fields

We study the Erdös–Falconer distance problem in vector spaces over finite fields with respect to the cubic metric. Estimates for discrete Airy sums and Adolphson–Sperber estimates for exponential sums in terms of Newton polyhedra play a crucial role. Similar techniques are used to study the incidence problem between points and cubic and quadratic curves. As a result we obtain a nontrivial range of exponents that appear to be difficult to attain using combinatorial methods.

Modified constructions of binary sequences using multiplicative inverse

Two new families of finite binary sequences are constructed using multiplicative inverse. The sequences are shown to have strong pseudorandom properties by using some estimates of certain exponential sums over finite fields. The constructions can be implemented fast since multiplicative inverse over finite fields can be computed in polynomial time.

On the exceptional set in Goldbach’s problem in short intervals

Let 1/5 < θ ≤ 1. We prove that there exists a positive constant δ such that the number of even integers in the interval [X, X + Xθ] which are not a sum of two primes is 《 Xθ−δ. The proof uses the circle method, a sieve method, exponential sum estimates and zero-density estimates for L-functions.

A slight improvement to Garaev's sum product estimate

max(\A + A\,\AA\)>Ce\A\?-*. In the finite field setting this situation is much more complicated because the main tool, the Szemer?di-Trotter incidence theorem, does not hold in the same generality. It is known, via the work in [BKT], that if A is a subset of Fp, the field of p elements with p prime, and if p6 0, then one has the sum product estimate max(|? + A|,|AA|)>|,4|1+ for some e > 0. This result has found many applications in combinatorial problems and exponential sum estimates (see e.g. [BKT], [BGK], [G2]). Recently, Garaev [Gl] showed that when \A < pz, one has the estimate

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

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.

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.

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).