The sequence \(x/n\) and its subsequences
type: Item
instance of: scholarly article
title: The sequence \(x/n\) and its subsequences
zbMATH Open document ID: 0881.11060
author: Roger C. Baker
author: Glyn Harman
published in: Rocky Mountain Journal of Mathematics
publication date: 1998-02-26T00:00:00Z
full work available at URL: contents.htm
review text: A positive integer \(n\) is said to be sparsely totient if \(\varphi(n)< \varphi(m)\) for all integers \(m>n\), where \(\varphi\) is Euler's function. One of the interesting problems concerning sparsely totient numbers \(n\) is to find the smallest number \(\lambda>0\) such that \[ \max_{p\mid n}p= O_\varepsilon ((\log n)^{\lambda+\varepsilon}), \] for all such \(n\), where \(p\) denotes a prime and \(\varepsilon>0\) is arbitrary. In this paper the authors prove that \(\lambda<37/20\). Since both the above problem and a problem on the Thue equation are linked to the behaviour of the sequences of the form \(x/r\) for given real \(x\), and natural numbers \(r\) in a suitable interval, in this paper, the authors also investigate a more general problem on these sequences. They prove the following result by a method which is adapted from \textit{W. Schmidt}'s beautiful work [in Small fractional parts of polynomials, Regional Conf. Ser. Math. 32, Am. Math. Soc. (1977; Zbl 0362.10032)]. Let \(R\) be a natural number, \({\mathbf x}\in\mathbb{R}^t\) and \(|{\mathbf x}|\leq R^{c_1}\). Let \(Q\) be a natural number with \(1<c_2\leq Q\leq R^{c_3}\) and let \(\psi_1,\dots, \psi_t\) be positive numbers satisfying \(\psi_j\leq c_4(\log Q)^{-t}\) for \(j=1,\dots,t\), and \(\psi_1\dots\psi_t= Q^{-1}\). Then \[ |x_j/r|\leq \psi_j, \qquad j=1,\dots, t \] for some integer \(r\) satisfying \(R\leq r\leq 2QR\), where \(|\alpha|\) denotes the distance from the real \(\alpha\) to the nearest integer. Here \(c_j\) are positive effectively computable constants, \(c_3\) depends on \(c_1\) and \(c_2\) and \(c_4\) depend on \(c_1\) and \(t\).
Mathematics Subject Classification ID: 11J54
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 11J13
zbMATH DE Number: 1026389
zbMATH Keywords: simultaneous approximation
zbMATH Keywords: small fractional parts
zbMATH Keywords: sparsely totient numbers
reviewed by: Ming-Chit Liu
MaRDI profile type: Publication
OpenAlex ID: W2071222125
cites work: Q4050665
cites work: The greatest prime factor of the integers in an interval
cites work: Small remainder of a vector to suitable modulus
cites work: Numbers with a large prime factor
cites work: Primes of the form \([n^ c]\)
cites work: On a problem of Erdős and Szemerédi
cites work: Q4033105
cites work: Q4161771
cites work: Q4195033
cites work: Q3900124
cites work: On the difference of consecutive terms of sequences defined by divisibility properties
cites work: On Gaps Between Squarefree Numbers II
cites work: Q3487501
cites work: Exponential sums with monomials
cites work: Q4179084
cites work: The Greatest Prime Factor of the Integers in an Interval
cites work: Q4692762
cites work: Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients
cites work: Q3910608
cites work: Q3720680
cites work: On the Distribution of αp; Modulo One
cites work: On sparsely totient numbers
cites work: On the Fractional Parts of n/j,j = o(n)
cites work: A new form of the error term in the linear sieve
cites work: \(P_ 2\) in short intervals
cites work: Q3825159
cites work: Q4101916
cites work: A Note on Numbers with a Large Prime Factor
cites work: Q5622297
cites work: On the fractional parts of \(x/n\) and related sequences. I
cites work: On the fractional parts of {x/n} and related sequences. II, III
cites work: On divisors of binomial coefficients. I
cites work: Fractional Parts of Pairs of Quadratic Polynomials
cites work: Q4135657
cites work: Diophantine approximation
cites work: On the difference of consecutive terms of sequences defined by divisibility properties, II
cites work: Nombres
DOI: 10.1216/RMJM/1181072023
description: scientific article; zbMATH DE number 1026389