Related papers: Nair-Tenenbaum bounds uniform with respect to the …
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…
The paper proves that a bound on the averaged Jones' square function of a measure implies an upper bound on the measure. Various types of assumptions on the measure are considered. The theorem is a generalization of a result due to A. Naber…
In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due…
A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as…
We consider the upper bound of Piltz divisor problem over number fields. Piltz divisor problem is known as a generalization of the Dirichlet divisor problem. We deal with this problem over number fields and improve the error term of this…
For a general class of non-negative functions defined on integral ideals of number fields, upper bounds are established for their average over the values of certain principal ideals that are associated to irreducible binary forms with…
For any quantity of interest in a system governed by ordinary differential equations, it is natural to seek the largest (or smallest) long-time average among solution trajectories, as well as the extremal trajectories themselves. Upper…
The more then hundred years old Bernstein inequality states that the supremum norm of the derivative of a trigonometric polynomial of fixed degree can be bounded from above by supremum norm of the polynomial itself. The reversed Bernstein…
In the late eighties, Hildebrand and Tenenbaum proved an asymptotic formula for the number of positive integers below $x$, having exactly $\nu$ distinct prime divisors: $\pi_{\nu}(x) \sim x \delta_{\nu}(x)$. Here we consider the restricted…
Given a suitable arithmetic function h, we investigate the average order of h as it ranges over the values taken by an integral binary form F. A general upper bound is obtained for this quantity, in which the dependence upon the…
This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely…
We provide an index bound for character sums of polynomials over finite fields. This improves the Weil bound for high degree polynomials with small indices, as well as polynomials with large indices that are generated by cyclotomic mappings…
In a previous article the authors determined the best-known upper bound for the cardinality of the image set for several classes of functions, including planar functions. Here, we show that the upper bound cannot be tight for planar…
We prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a…
An important tool to quantify the likeness of two probability measures are f-divergences, which have seen widespread application in statistics and information theory. An example is the total variation, which plays an exceptional role among…
We address the problem of producing a lower bound for the mean of a discrete probability distribution, with known support over a finite set of real numbers, from an iid sample of that distribution. Up to a constant, this is equivalent to…
By relating the number of images of a function with finite domain to a certain parameter, we obtain both an upper and lower bound for the image set. Even though the arguments are elementary, the bounds are, in some sense, best possible. The…
We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This…
This paper develops sharp bounds on moments of sums of k-wise independent bounded random variables, under constrained average variance. The result closes the problem addressed in part in the previous works of Schmidt et al. and Bellare,…
We give a distribution-dependent concentration inequality for functions of independent variables. The result extends Bernstein's inequality from sums to more general functions, whose variation in any argument does not depend too much on the…