Exponential Sums Research Articles

On restriction of exponential sums to hypersurfaces with zero curvature

We prove essentially sharp bounds for the $$L^p$$ restriction of weighted Gauss sums to monomial curves. Getting the $$L^2$$ upper bound combines the $$TT^*$$ method for matrices with the first and second derivative test for exponential sums. The matching lower bound follows via constructive interference on short blocks of integers, near the critical point of the phase function. This method is used to make the broader point that restriction to hypersurfaces is really sensitive to curvature. Our results here complement those in [4].

A fast finite volume method for spatial fractional diffusion equations on nonuniform meshes

In this paper, a fast finite volume method is proposed for the initial and boundary value problems of spatial fractional diffusion equations on nonuniform meshes. The discretizations of the Riemann-Liouville fractional derivatives lead to unstructured dense coefficient matrices, differing from the Toeplitz-like structure under the uniform mesh. The fast algorithm is proposed by using the sum-of-exponentials (SOE) technique to the spatial kernel xα−1,α∈(0,1). Then, the matrix-vector multiplications of the resulting coefficient matrices could be implemented in O(mlog2⁡m) operations, where m denotes the size of matrices. Iterative solvers are preferably applied to obtain the numerical solution. The proposed fast scheme is proved to be unconditionally stable for sufficiently accurate SOE approximation. Meanwhile, a banded preconditioner is exploited to accelerate the Krylov subspace method. Numerical experiments are provided to demonstrate the efficiency of the proposed fast algorithm.

On multiplicative energy of subsets of varieties

AbstractWe obtain a nontrivial upper bound for the multiplicative energy of any sufficiently large subset of a subvariety of a finite algebraic group. We also find some applications of our results to the growth of conjugates classes, estimates of exponential sums, and restriction phenomenon.

Some methods for solving equations with an operator function and applications for problems with a fractional power of an operator

Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and numerical methods for their study have been actively discussed. Computational algorithms for such non-standard problems are based on approximations by the operator function. In this case, an approximate solution is determined by solving auxiliary standard problems. This paper discusses the main directions for constructing acceptable approximations of operator functions when solving equations.The most widespread are the approaches using various options for rational approximation. Also, we note the methods that relate to approximation by exponential sums. We propose to use a new approach, which is based on the application of approximation by exponential products. The solution of an equation with an operator function is based on the transition to standard stationary or evolutionary problems. Estimates of the accuracy of the approximate solution of the operator equation at known absolute or relative errors of approximation functions are obtained. The influence of the error in the solution of auxiliary problems is investigated separately.General approaches are illustrated by a problem with a fractional power of the operator. The first class of methods is based on the integral representation of the operator function under rational approximation, approximation by exponential sums, and approximation by exponential products. The second class of methods is associated with solving an auxiliary Cauchy problem for some evolutionary equation. In this case, we can distinguish a method based on approximation by exponential products.

Metric theory of Weyl sums

Metric theory of Weyl sums

Approximating the Gaussian as a Sum of Exponentials and Its Applications to the Fast Gauss Transform

Approximating the Gaussian as a Sum of Exponentials and Its Applications to the Fast Gauss Transform

Efficient Scaling of Window Function Expressed as Sum of Exponentials

Efficient Scaling of Window Function Expressed as Sum of Exponentials

Markov chains on finite fields with deterministic jumps

We study the Markov chain on Fp obtained by applying a function f and adding ±γ with equal probability. When f is a linear function, this is the well-studied Chung–Diaconis–Graham process. We consider two cases: when f is the extension of a non-linear rational function which is bijective, and when f(x)=x2. In the latter case, the stationary distribution is not uniform and we characterize it when p=3(mod4). In both cases, we give an almost linear bound on the mixing time, showing that the deterministic function dramatically speeds up mixing. The proofs involve establishing bounds on exponential sums over the union of short intervals.

Ternary perfect sequences with three-valued cross-correlation

<p style='text-indent:20px;'>The calculation of the cross-correlation between a sequence with good autocorrelation and its decimated sequence has been a longstanding research problem in the field of sequence design. The objective of this paper is to determine the cross-correlation between a class of well-known ternary sequences with perfect autocorrelation and its <inline-formula><tex-math id="M1">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-decimation. Based on the theory of quadratic forms and exponential sums over finite fields, it is shown that the cross-correlation function takes on three low values.</p>

Interference Mitigation for FMCW Radar With Sparse and Low-Rank Hankel Matrix Decomposition

In this paper, the interference mitigation for Frequency Modulated Continuous Wave (FMCW) radar system with a dechirping receiver is investigated. After dechirping operation, the scattered signals from targets result in beat signals, i.e., the sum of complex exponentials while the interferences lead to chirp-like short pulses. Taking advantage of these different time and frequency features between the useful signals and the interferences, the interference mitigation is formulated as an optimization problem: a sparse and low-rank decomposition of a Hankel matrix constructed by lifting the measurements. Then, an iterative optimization algorithm is proposed to tackle it by exploiting the Alternating Direction of Multipliers (ADMM) scheme. Compared to the existing methods, the proposed approach does not need to detect the interference and also improves the estimation accuracy of the separated useful signals. Both numerical simulations with point-like targets and experiment results with distributed targets (i.e., raindrops) are presented to demonstrate and verify its performance. The results show that the proposed approach is generally applicable for interference mitigation in both stationary and moving target scenarios.

Binary sequences and lattices constructed by discrete logarithms

<abstract><p>In 1997, Mauduit and Sárközy first introduced the measures of pseudorandomness for binary sequences. Since then, many pseudorandom binary sequences have been constructed and studied. In particular, Gyarmati presented a large family of pseudorandom binary sequences using the discrete logarithms. Ten years later, to satisfy the requirement from many applications in cryptography (e.g., in encrypting "bit-maps'' and watermarking), the definition of binary sequences is extended from one dimension to several dimensions by Hubert, Mauduit and Sárközy. They introduced the measure of pseudorandomness for this kind of several-dimension binary sequence which is called binary lattices. In this paper, large families of pseudorandom binary sequences and binary lattices are constructed by both discrete logarithms and multiplicative inverse modulo $ p $. The upper estimates of their pseudorandom measures are based on estimates of either character sums or mixed exponential sums.</p></abstract>

The fourth power mean value of one kind two-term exponential sums

<abstract><p>In this paper, based on the analytic method and the properties of Gauss sums, we study the computational problems of the fourth power mean value of one kind two-term exponential sums through the classification and estimation of Dirichlet characters and give it a calculation formula or asymptotic formula in different conditions.</p></abstract>

One Kind New Hybrid Power Mean and Its Computational Formulae

The main purpose of this study is to use the elementary and analytic methods and the properties of the classical Gauss sums to study the calculation problems of one kind of hybrid power mean involving the quadratic character sums and the two‐term exponential sums and give an exact computational formula for it.

On smooth Weyl sums over biquadrates and Waring’s problem

We provide estimates for $s$th moments of biquadratic smooth Weyl sums, when $10\le s\le 12$, by enhancing the second author’s iterative method that delivers estimates beyond the classical convexity barrier. As a consequence, all sufficiently large intege

The Maximal Difference of Different Powers of an Element Modulo n

In this paper, we investigate the maximal difference of integer powers of an element modulo n . Let a n denote the integer b with 1 ≤ b ≤ n such that a ≡ b mod n for any integer a . Using the bounds for exponential sums, we obtain a lower bound of the function H m 1 , m 2 n : = max a m 1 n − a m 2 n : 1 ≤ a ≤ n , a , n = 1 , which gives n − H m 1 , m 2 n = O n 3 / 4 + o 1 .

The Vortex-like Behavior of the Riemann Zeta Function to the Right of the Critical Strip

Based on an equivalence relation that was established recently on exponential sums, in this paper we study the class of functions that are equivalent to the Riemann zeta function in the half-plane {sin {mathbb {C}}:mathrm{Re}, s>1}. In connection with this class of functions, we first determine the value of the maximum abscissa from which the images of any function in it cannot take a prefixed argument. The main result shows that each of these functions experiments a vortex-like behavior in the sense that the main argument of its images varies indefinitely near the vertical line mathrm{Re}, s=1. In particular, regarding the Riemann zeta function zeta (s), for every sigma _0>1 we can assure the existence of a relatively dense set of real numbers {t_m}_{mge 1} such that the parametrized curve traced by the points (mathrm{Re} (zeta (sigma +it_m)),mathrm{Im}(zeta (sigma +it_m))), with sigma in (1,sigma _0), makes a prefixed finite number of turns around the origin.

On digits of Mersenne numbers

Motivated by recently developed interest to the distribution of q -ary digits of Mersenne numbers M_p = 2^p-1 , where p is prime, we estimate rational exponential sums with M_p , p \le X , modulo a large power of a fixed odd prime q . In turn this immediately implies the normality of strings of q -ary digits amongst about (\log X)^{3/2+o(1)} rightmost digits of M_p , p \le X . Previous results imply this only for about (\log X)^{1+o(1)} rightmost digits.

Generic Newton polygons for L-functions of (A,B)-exponential sums

In this paper, we consider the following (A,B)-polynomial f over finite field:f(x0,x1,⋯,xn)=x0Ah(x1,⋯,xn)+g(x1,⋯,xn)+PB(1/x0), where h is a Deligne polynomial of degree d, g is an arbitrary polynomial of degree less than dB/(A+B) and PB(y) is a one-variable polynomial of degree less than or equal to B. Let Δ be the Newton polyhedron of f at infinity. We show that Δ is generically ordinary if p≡1modD, where D is a constant determined by Δ. In other words, we prove that the Adolphson–Sperber conjecture is true for Δ.

Rational Functions for the Reconstruction of Exponential Sums from their Fourier Coefficients

AbstractWe look closer at the type of rational functions involved in reconstructing exponential sums from their Fourier coefficients.

Dynamical invariants and intersection theory on the flex and gothic loci

The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points. The flex and gothic loci provide the first example of a primitive, totally geodesic subvariety of moduli space and new \({\text {SL}}_2({\mathbb {R}})\)-invariant varieties in Teichmüller dynamics, as discovered by McMullen–Mukamel–Wright. In this paper we determine the divisor class of the flex locus as well as various tautological intersection numbers on the gothic locus. For the case of the gothic locus our result confirms numerically a conjecture of Chen–Möller–Sauvaget about computing sums of Lyapunov exponents for \({\text {SL}}_2({\mathbb {R}})\)-invariant varieties via intersection theory.