Goldbach’s conjecture in arithmetic progressions: number and size of exceptional prime moduli
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DOI: 10.1007/s00013-016-0993-0
Abstract: Set $${T=N^{\frac{1}{3}-\epsilon}}$$T=N13-ϵ. It is proved that for all but $${\ll TL^{-H},\,H > 0}$$≪TL-H,H>0, exceptional prime numbers $${k\leq T}$$k≤T and almost all integers b1, b2 co-prime to k, almost all integers $${n\sim N}$$n∼N satisfying $${n\equiv b_{1}+b_{2}(mod\,k)}$$n≡b1+b2(modk)… read more here.
Keywords: progressions number; number size; conjecture arithmetic; exceptional prime ... See more keywords
Arithmetic Progressions and Chaos in Linear Dynamics
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DOI: 10.1007/s00020-022-02687-3
Abstract: We characterize chaotic linear operators on reflexive Banach spaces in terms of the existence of long arithmetic progressions in the sets of return times. To achieve this, we study $\mathcal F$-hypercyclicity for two families of… read more here.
Keywords: chaos linear; progressions chaos; linear dynamics; arithmetic progressions ... See more keywords
Szemerédi’s proof of Szemerédi’s theorem
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DOI: 10.1007/s10474-020-01081-y
Abstract: In 1975, Szemeredi famously established that any set of integers of positive upper density contained arbitrarily long arithmetic progressions. The proof was extremely intricate but elementary, with the main tools needed being the van der… read more here.
Keywords: theorem; proof; szemer proof; regularity lemma ... See more keywords
Powers in arithmetic progressions
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DOI: 10.1007/s11139-020-00331-5
Abstract: We investigate the function $$P_{a,b;N}(\ell )$$ describing the number of $$\ell $$ -th powers among the first N terms of an arithmetic progression $$ax+b$$ . We completely describe the arithmetic progressions containing the most $$\ell… read more here.
Keywords: powers arithmetic; arithmetic progressions;
Arithmetic progressions in self-similar sets
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DOI: 10.1007/s11464-019-0788-2
Abstract: Given a sequence $\{b_{i}\}_{i=1}^{n}$ and a ratio $\lambda \in (0,1),$ let $E=\cup_{i=1}^n(\lambda E+b_i)$ be a homogeneous self-similar set. In this paper, we study the existence and maximal length of arithmetic progressions in $E$. Our main… read more here.
Keywords: self similar; similar sets; progressions self; arithmetic progressions ... See more keywords
On arithmetic progressions in Lucas sequences – II
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DOI: 10.1016/j.jnt.2019.02.015
Abstract: Abstract In this paper, we provide an effective and practical method to find all three term arithmetic progressions in a given Lucas sequence of the first or second kind. Our interest is the case when… read more here.
Keywords: progressions lucas; lucas sequences; arithmetic progressions;
Generalized divisor functions in arithmetic progressions: I
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DOI: 10.1016/j.jnt.2021.03.021
Abstract: Abstract We prove some distribution results for the k-fold divisor function in arithmetic progressions to moduli that exceed the square-root of length X of the sum, with appropriate constrains and averaging on the moduli, saving… read more here.
Keywords: generalized divisor; divisor functions; arithmetic progressions; functions arithmetic ... See more keywords
Squarefree Integers in Arithmetic Progressions to Smooth Moduli
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DOI: 10.1017/fms.2021.67
Abstract: Abstract Let $\varepsilon> 0$ be sufficiently small and let $0 < \eta < 1/522$ . We show that if X is large enough in terms of $\varepsilon $ , then for any squarefree integer $q… read more here.
Keywords: squarefree integers; smooth moduli; progressions smooth; power ... See more keywords
A characterization of finite arithmetic progressions via regular partitions of quadrilaterals
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DOI: 10.1080/0020739x.2019.1671994
Abstract: ABSTRACT In this article, we will give a geometric interpretation of certain finite arithmetic progressions. For this purpose, we will introduce the concept of the n-regular partition of a quadrilateral. read more here.
Keywords: arithmetic progressions; via regular; progressions via; regular partitions ... See more keywords
MEAN VALUES OF ARITHMETIC FUNCTIONS IN SHORT INTERVALS AND IN ARITHMETIC PROGRESSIONS IN THE LARGE‐DEGREE LIMIT
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DOI: 10.1112/mtk.12032
Abstract: 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… read more here.
Keywords: large degree; arithmetic progressions; degree limit; interval ... See more keywords
SET MAPPINGS WITH FREE SETS WHICH ARE ARITHMETIC PROGRESSIONS
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DOI: 10.1112/s0025579317000353
Abstract: If 3 ≤ n < ω, V is a vector space over Q with |V | ≤ אn−2, then there is a well ordering of V such that every vector is the last element of… read more here.
Keywords: sets arithmetic; free sets; set mappings; mappings free ... See more keywords