相关论文: Diophantine methods for exponential sums, and expo…
In this preprint we consider generalizations of discrete and integral Cauchy--Bunyakovskii inequalities by the method of mean values with some applications. Mostly the material is compiled as a short survey but some results are proved. Main…
In this article, we introduce and study three numerical methods for the Dirichlet Monge Amp\`ere equation in two dimensions. The approaches consist in considering new equivalent problems. The latter are discretized by a wide stencil finite…
In these notes we discuss some relations between complex analysis (derivatives of Cauchy integrals) and curvatures of curves and surfaces. In higher dimensions the Cauchy integrals are based on generalizations of complex analysis using…
In this paper we study the Diophantine equation \begin{align*} b^k + \left(a+b\right)^k + &\left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) + b\right)^k = \\ &y\left(y+c\right) \left(y+2c\right) \ldots \left(y+…
We estimate short exponential sums weighted by the Fourier coefficients of a Maass form. This requires working out a certain transformation formula for non-linear exponential sums, which is of independent interest. We also discuss how the…
First we generalize a famous lemma of Gallagher on the mean square estimate for exponential sums by plugging a weight in the right hand side of Gallagher's original inequality. Then we apply it in the special case of the Cesaro weight, in…
We discuss some classical and recent results and open problems on the statistical behavior of ergodic sums above toral translations, and their applications to Diophantine approximations and to ergodic properties of systems related to…
In the first part of this expository paper, we present and discuss the interplay of Dirichlet polynomials in some classical problems of number theory, notably the Lindel\"of Hypothesis. We review some typical properties of their means and…
These are the notes of a part of the PhD course Regularity for free boundary problems and for elliptic PDEs, held in Pavia in the spring of 2025. The aim is to provide a comprehensive and self-contained treatment of classical interior and…
By combining classical techniques together with two novel asymptotic identities contained in [FL], we analyse certain single sums of Riemann-zeta type. In addition, we analyse Euler-Zagier double exponential sums for particular values of…
Recent progress in analytical calculation of the multiple [inverse, binomial, harmonic] sums, related with epsilon-expansion of the hypergeometric function of one variable are discussed.
We show that if an exponential sum with multiplicative coefficients is large then the associated multiplicative function is "pretentious". This leads to applications in the circle method, and a natural interpretation of the local-global…
In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarn\'{\i}k, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various…
We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in…
This paper is concerned with the study of the fractional finite sums theory. We present the classes of functions for which it is possible to characterize the constant related to the derivative of fractional sums (denominated by essence of a…
Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…
In 2004, J.C. Tong found bounds for the approximation quality of a regular continued fraction convergent of a rational number, expressed in bounds for both the previous and next approximation. We sharpen his results with a geometric method…
We exhibit differential geometric structures that arise in numerical methods, based on the construction of Cauchy sequences, that are currently used to prove explicitly the existence of weak solutions to functional equations. We describe…
Using elementary methods, we establish old and new relations between binomial coefficients, Fibonacci numbers, Lucas numbers, and more.
We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the Difference Potentials framework. The main novelty of the developed schemes is the use of local…