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Uniform distribution of polynomially-defined additive functions to varying moduli

INTERNATIONAL JOURNAL OF NUMBER THEORY 2025年第4期21卷 959-973页

作者： Akande, Agbolade Univ Georgia Chapel Herty Dr Athens GA 30602 USA

There have been results on uniform distribution modulo 1 of sequences of the form {alpha f(n)}(infinity)(n=1 )where f (n) is an arithmetic function and alpha is an irrational number. For example, {alpha n}(infinity)(n=1 ) (Bohl, Sierpinski and Weyl) and {alpha Omega( n ) } (infinity)(n=1 )(Erdos and Delange) have been shown to be uniformly distributed modulo 1 for all irrational numbers alpha . De Koninck and Katai have shown that {alpha phi(n)}(infinity)(n=1 ) and {alpha sigma(n)}(infinity)(n=1 ) are uniformly distributed modulo 1 for a subset of irrational numbers alpha . In this article, we will extend their result by showing that the sequences {alpha phi(n)}(infinity)(n=1 ) and {alpha sigma(n)}(infinity)(n=1 )are uniformly distributed modulo 1 when alpha is a non-Liouville number. The proof will use Weyl's criterion, upper bounds of exponential functions established by Vinogradov and Vaughan, and the notion of a thin set established by Pollack and Vandehey. There are two corollaries that arise from the result of this article: { 10 (alpha phi )( n ) } (infinity)(n=1 ) and { 10 alpha sigma ( n ) } (infinity)(n=1 )are strong Benford sequences for all non-Liouville numbers alpha , and the sequences {F(n) + alpha phi(n)}(infinity)(n=1 )and {F(n) + alpha sigma ( n ) } (infinity)(n=1 ) are uniformly distributed modulo 1 for all non-Liouville numbers alpha and additive function F .

关键词： Uniform distribution additive function exponential sums residue classes

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A class of functional equations for additive functions

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AEQUATIONES MATHEMATICAE 2025年第3期99卷 825-839页

作者： Ebanks, Bruce Mississippi State Univ Dept Math & Stat Mississippi State MS 39762 USA

The study of functional equations in which the unknown functions are assumed to be additive has a long history and continues to be an active area of research. Here we discuss methods for solving functional equations of the form (& lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}) & sum;j=1kxpjfj(xqj)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}<^>{k} x<^>{p_j}f_j(x<^>{q_j}) = 0$$\end{document}, where the pj,qj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_j,q_j$$\end{document} are non-negative integers, the fj:R -> S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_j:R \rightarrow S$$\end{document} are additive functions, S is a commutative ring, and R is a sub-ring of S. This area of research has ties to commutative algebra since homomorphisms and derivations satisfy equations of this type. Methods for solving all homogeneous equations of the form (& lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}) can be found in Ebanks (Aequ Math 89(3):685-718, 2015), Ebanks (Results Math 73(3):120, 2018) and Gselmann et

关键词： Ring additive function Derivation Higher-order derivation functional equation Homogeneity

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Cauchy-Schwarz-type inequalities for solutions of Levi-Civita-type functional equations

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PERIODICA MATHEMATICA HUNGARICA 2025年第2期90卷 228-238页

作者： Pales, Zsolt Shihab, Mahmood Kamil Univ Debrecen Inst Math Pf 400 H-4002 Debrecen Hungary Univ Debrecen Doctoral Sch Math & Computat Sci Debrecen Hungary Univ Kirkuk Coll Educ Pure Sci Dept Math Kirkuk Iraq

The main goal of this paper is to show that if a real-valued function defined on a groupoid satisfies a certain Levi-Civita-type functional equation, then it also fulfills a Cauchy-Schwarz-type functional inequality. In particular, if the groupoid is the multiplicative structure of a commutative ring, then we can establish the existence of nontrivial additive functions satisfying inequalities connected to the multiplicative structure.

关键词： Levi-Civita-type functional equation Cauchy-Schwarz-type functional inequality additive function

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A study on Davison's functional equation for set-valued functions

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AEQUATIONES MATHEMATICAE 2025年 1-10页

作者： Mohammadi, Elham Najati, Abbas EL-Fassi, Iz-iddine Univ Mohaghegh Ardabili Dept Math Ardebil Iran SM Ben Abdellah Univ Fac Sci & Tech Dept Math BP 2202 Fes Morocco

Let X be a unital algebra with unit e and Y a real Hausdorff topological vector space. In this article, we obtain the general set-valued solution of the Davison functional equation F(xy+x)+F(y)=F(xy)+F(x+y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(xy+x)+F(y)=F(xy)+F(x+y) \end{aligned}$$\end{document}for functions F defined on X with values in the set of nonempty compact and convex subsets of Y. Additionally, we investigate various extensions of the Davison set-valued functional equation.

关键词： Set-valued function additive function Davison functional equation

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A BOREL LINEAR SUBSPACE OF Rω THAT CANNOT BE COVERED BY COUNTABLY MANY CLOSED HAAR-MEAGER SETS

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TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS 2024年第1期63卷 197-208页

作者： Banakh, Taras Jablonska, Eliza Ivan Franko Univ Lviv Lvov Ukraine Jan Kochanowski Univ Kielce Kielce Poland AGH Univ Krakow Fac Appl Math Mickiewicza 30 PL-30059 Krakow Poland

We prove that the countable product of lines contains a Haarnull Haar -meager Borel linear subspace L that cannot be covered by countably many closed Haar -meager sets. This example is applied to studying the interplay between various classes of "large" sets and Kuczma-Ger classes in the topological vector spaces R n for n <= omega .

关键词： additive function mid-convex function continuity Haar-null set Haar-meager set null-finite set Haar-thin set Polish Abelian group Ger-Kuczma classes

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On Ramanujan expansions with additive coefficients

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INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS 2023年第2期56卷 1-6页

作者： Laporta, Maurizio Univ Napoli Complesso Monte S Angelo Dipartimento Matemat & Applicaz Via Cinthia I-80126 Naples NA Italy

The Ramanujan series attached to a complex-valued arithmetic function g ($) over cap in a fixed integer a is the series Sigma(n)g ($) over cap (n)cn(a), where cn(a) is the so-calledRamanujan sum. Assuming that g ($) over cap is additive or, more generally, a product of a multiplicative function with an additive one, we study the relationships between the Ramanujan series attached to g ($) over cap in a positive integer a and its subseries obtained by taking the terms with n coprime to a fixed integer d >= 2.

关键词： Ramanujan expansion additive function Multiplicative function Series identity

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On Numbers Not Representable as n plus w(n)

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MATHEMATICAL NOTES 2023年第3-4期113卷 384-395页

作者： Kucheryavyi, P. A. Natl Res Univ Higher Sch Econ Fac Math Moscow 101000 Russia

w(n) be an additive nonnegative integer-valued arithmetic function equal to 1 on primes. We study the distribution of n + w(n) modulo a prime p and give a lower bound for the density of numbers not representable as n ... 详细信息

关键词： number of prime divisors Perron's formula additive function

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Basic sets for some basic functional equations

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AEQUATIONES MATHEMATICAE 2023年第5-6期97卷 1221-1230页

作者： Ger, Roman Univ Silesia Inst Math Bankowa 14 PL-40007 Katowice Poland

Motivated by the well known fact that any nonzero solution of the fundamental Cauchy functional equation may arbitrarily be prescribed on a Hamel basis, we deal with the following problem: given a functional equation E-1(phi) = E2(phi) (*) with the unknown function phi : X -> Y, what must the set (SIC) (SIC) = Z subset of X be like in order to ensure that an arbitrary *** 0 : Z -> Y admits a unique function phi : X -> Y solving equation (*) and such that phi(| Z) =phi 0;if such a set does exist it is termed to be a basic set. We discuss the problem of existence of basic sets for exponential functions, d'Alembert's functions, sine functions, Cuculi`ere's functions and hyperbolic tangent type functions, among others.

关键词： Basic set Addition formula functional equation additive function Exponential function Alternative (conditional) functional equations D'Alembert's equation Sine equation

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Interdependence of additivity and sine additivity

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AFRIKA MATEMATIKA 2024年第2期35卷 1-21页

作者： Ebanks, Bruce Mississippi State Univ Dept Math & Stat Mississippi State MS 39762 USA

Let S be a semigroup and K a field. A function f:S -> K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:S \rightarrow K$$\end{document} is additive if f(xy)=f(x)+f(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(xy) = f(x) + f(y)$$\end{document} for all x,y is an element of S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y \in S$$\end{document}, and functions g,h:S -> K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g,h:S \rightarrow K$$\end{document} form a sine pair if they satisfy the sine addition law g(xy)=g(x)h(y)+h(x)g(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(xy) = g(x)h(y) + h(x)g(y)$$\end{document} for all x,y is an element of S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y \in S$$\end{document}. Adding these two equations we arrive at the functional equation (*) f(xy)+g(xy)=f(x)+f(y)+g(x)h(y)+h(x)g(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage

关键词： Alienation functional equation additive function Sine addition law Semigroup

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Stability of an Abel functional equation over restricted domains

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RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO 2024年第8期73卷 2913-2919页

作者： Mohammadi, Elham Najati, Abbas EL-Fassi, Iz-iddine Univ Mohaghegh Ardabili Dept Math Fac Sci Ardebil Iran SM Ben Abdellah Univ Dept Math Fac Sci & Tech BP 2202 Fes Morocco

Let R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R$$\end{document} be the set of real numbers and Y a Banach space. In this work, we study the Hyers-Ulam stability for the Abel functional equation f(x+y)=g(xy)+h(x-y),x,y is an element of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(x+y)=g(xy)+h(x-y),\;\;x,y\in \mathbb R \end{aligned}$$\end{document}where f,g,h:R -> Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f,g,h:\mathbb R\rightarrow Y$$\end{document} are unknown functions. The perturbation of above equation in the restricted domains are also proved. As a direct consequence we obtain asymptotic behaviors of the proposed equation.

关键词： Abel functional equation additive function Asymptotic behavior

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