Estimates For Sums Research Articles

Non-asymptotic Results for Singular Values of Gaussian Matrix Products

This article provides a non-asymptotic analysis of the singular values (and Lyapunov exponents) of Gaussian matrix products in the regime where N, the number of terms in the product, is large and n, the size of the matrices, may be large or small and may depend on N. We obtain concentration estimates for sums of Lyapunov exponents, a quantitative rate for convergence of the empirical measure of the squared singular values to the uniform distribution on [0, 1], and results on the joint normality of Lyapunov exponents when N is sufficiently large as a function of n. Our technique consists of non-asymptotic versions of the ergodic theory approach at \(N=\infty \) due originally to Furstenberg and Kesten (Ann Math Stat 31(2):457–469, 1960) in the 1960s, which were then further developed by Newman (Commun Math Phys 103(1):121–126, 1986) and Isopi and Newman (Commun Math Phys 143(3):591–598, 1992) as well as by a number of other authors in the 1980s. Our key technical idea is that small ball probabilities for volumes of random projections gives a way to quantify convergence in the multiplicative ergodic theorem for random matrices.

Square-root cancellation for sums of factorization functions over short intervals in function fields

We present new estimates for sums of the divisor function and other similar arithmetic functions in short intervals over function fields. (When the intervals are long, one obtains a good estimate from the Riemann hypothesis.) We obtain an estimate that approaches square-root cancellation as long as the characteristic of the finite field is relatively large. This is done by a geometric method, inspired by work of Hast and Matei, where we calculate the singular locus of a variety whose Fq-points control this sum. This has applications to highly unbalanced moments of L-functions.

P-adic estimates of exponential sums on curves

The purpose of this article is to prove a ``Newton over Hodge'' result for exponential sums on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. For a regular function $\overline{f}$ on $V$, we may form the $L$-function $L(\overline{f},V,s)$ associated to the exponential sums of $\overline{f}$. In this article, we prove a lower estimate on the Newton polygon of $L(\overline{f},V,s)$. The estimate depends on the local monodromy of $f$ around each point $x \in X-V$. This confirms a hope of Deligne that the irregular Hodge filtration forces bounds on $p$-adic valuations of Frobenius eigenvalues. As a corollary, we obtain a lower estimate on the Newton polygon of a curve with an action of $\mathbb{Z}/p\mathbb{Z}$ in terms of local monodromy invariants.

Development of a weighted sum estimate of the total radiated power from large helical device plasma.

Diagnosing the amount of radiated power is an important research goal for fusion devices. This research aims at better understanding and diagnosing the radiated power from the Large Helical Device (LHD). The current radiated power estimate in the LHD is based on one wide-angle resistive bolometer. Because the estimate stems from one bolometer location toroidally and has a wide-angle poloidal view, this estimate does not take into account toroidal and poloidal radiation asymmetries that are observed in the LHD in discharges with gas puffing. This research develops a method based on the EMC3-Eirene model to calculate the set of coefficients for a weighted-sum method of estimating the radiated power. This study calculates these coefficients by using a least-squares method to solve for a coefficient set, using a variety of simulated cases generated by the EMC3-Eirene model, combined with corresponding geometric radiated power density considerations. If this set of coefficients is multiplied by the detector signal of each bolometer and summed up, this gives a total radiated power estimate. This new estimate takes into account toroidal and poloidal asymmetries by using the bolometer channels viewing different toroidal and poloidal locations, thereby reducing the estimation error and providing information about toroidal asymmetries.

Estimates for Weyl sums in the theory of discrete universality

We prove joint and disjoint discrete universality theorems for Dirichlet L-functions $$L(s,\chi )$$ and Hurwitz zeta-functions $$\zeta (s;\beta )$$ with rational parameter $$\beta \in (0,1]$$ . Our approach does not utilize Gallagher’s lemma, which is usually employed to prove discrete universality theorems. In our case, however, this would lead to certain difficulties when it is about estimating discrete second moments of L-functions. Therefore, we introduce a novel approach which is based only on Euler product representations and zero-density estimates of L-functions, as well as mean-value estimates for Weyl sums.

On the dynamical system generated by the Möbius transformation at prime times

We study the distribution of the sequence of elements of the discrete dynamical system generated by iterations of the Möbius map \(x \mapsto (ax + b)/(cx+d)\) over a finite field of p elements at the moments of time that correspond to prime numbers. In particular, we obtain nontrivial estimates of exponential sums with such sequences.

Sums of free variables in fully symmetric spaces

We give a method to obtain, from Voiculescu's inequality, norm estimates for sums of free variables with amalgamation in general fully symmetric spaces. We use these estimates to interpolate the Burkholder inequalities for non-commutative martingales. The method is also applicable to other similar settings. In that spirit, we improve known results on the non-commutative Johnson–Schechtman inequalities and recover Khinchin inequalities associated to free groups.

Higher order moments of generalized quadratic Gauss sums weighted by $L$-functions

The main purpose of this paper is to study higher order moments of the generalized quadratic Gauss sums weighted by $L$-functions using estimates for character sums and analytic methods. We find asymptotic formulas for three character sums which arise naturally in the study of higher order moments of the generalized quadratic Gauss sums. We then use these character sum estimates to find asymptotic formulas for the $\nth{6}$ and $\nth{8}$ order moments of the generalized quadratic Gauss sums weighted by $L$-functions. Our asymptotic formulas satisfy a conjecture of Wenpeng Zhang.

Estimates of short exponential sums with primes in major arcs

Estimates of short exponential sums with primes in major arcs

The remark on the mean value theorem for the absolute value of Dirichlet L-function in the critical strip

We continue our researches concerning the generalization and improvement of R.T.Turganaliev’s result that states an asymptotic formula for the mean value of the Riemann zeta function in the critical strip with power factor saving in the remainder term.We find an asymptotic for the mean value of Dirichlet L-function in the critical strip. This assertion improves R.T.Turganaliev’s theorem for zeta-function in the whole interval $$(1/2 < Re 𝑠 ≤ 1)$$. Our result is based on the special use of the estimation of exponential sums by second derivative test.

Secretary problem with vanishing objects

We consider a version of the secretary problem where elements may vanish during the selection and become unchoosable. We construct a selection strategy and identify the probability to select the best element, which turns out to be asymptotically maximal as number of elements increases indefinitely. As an auxiliary result of independent interest we establish large deviation probability estimates for sums of independent variables with distinct geometric distribution.

Bounds for convergence rate in laws of large numbers for mixed Poisson random sums

In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev ζ-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous Rényi theorem) to a considerably wider class of random indices with mixed Poisson distributions including, e.g., those with the (generalized) negative binomial distribution.

A note on character sums in function fields

We give an improvement on a character sum estimate in Fq[t] and answer a question of Shparlinski.

Infinite Paley graphs

Infinite analogues of the Paley graphs are constructed, based on uncountably many locally finite fields. By using character sum estimates due to Weil, they are shown to be isomorphic to the countable random graph of Erdős, Rényi and Rado.

Small cap decouplings

We develop a toolbox for proving decouplings into boxes with diameter smaller than the canonical scale. As an application of this new technique, we solve three problems for which earlier methods have failed. We start by verifying the small cap decoupling for the parabola. Then we find sharp estimates for exponential sums with small frequency separation on the moment curve in $$\mathbb {R}^3$$ . This part of the work relies on recent improved Kakeya-type estimates for planar tubes, as well as on new multilinear incidence bounds for plates and planks. We also combine our method with the recent advance on the reverse square function estimate, in order to prove small cap decoupling into square-like caps for the two dimensional cone. The Appendix by Roger Heath-Brown contains an application of the new exponential sum estimates for the moment curve, to the Riemann zeta-function.

Hua’s theorem on five squares of primes

We give an alternative proof of Hua’s theorem that each large N ≡ 5 (mod 24) can be written as a sum of five squares of primes. The proof depends on an estimate of exponential sums involving the Mobius function.

Mean Values of Arithmetic Functions under Congruences with the Euler Function

We examine the average order of some arithmetic functions written as sums over Euler function in arithmetic progression and in general over such that is a prime number, an integer and is a polynomial function with integer coefficients and a degree that is not constant modulo Our results are based on various estimates of rational exponential sums with the Euler Function in arithmetic progression which are due to William Banks and Igor E. Shparlinski.

Uniform L p Resolvent Estimates on the Torus

A new range of uniform L p resolvent estimates is obtained in the setting of the flat torus, improving previous results of Bourgain, Shao, Sogge and Yao. The arguments rely on the ℓ 2 -decoupling theorem and multidimensional Weyl sum estimates.

Strichartz estimates for the Schrödinger flow on compact Lie groups

We establish scale-invariant Strichartz estimates for the Schr\"odinger flow on any compact Lie group equipped with canonical rational metrics. In particular, full Strichartz estimates without loss for some non-rectangular tori are given. The highlights of this paper include estimates for some Weyl type sums defined on rational lattices, different decompositions of the Schr\"odinger kernel that accommodate different positions of the variable inside the maximal torus relative to the cell walls, and an application of the BGG-Demazure operators or Harish-Chandra's integral formula to the estimate of the difference between characters.

Estimates for sums and gaps of eigenvalues of Laplacians on measure spaces

AbstractFor Laplacians defined by measures on a bounded domain in ℝn, we prove analogues of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, Pólya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.