Open Access
February, 2022 Maximal Density of Integral Sets with Missing Differences and the Kappa Values
Ram Krishna Pandey, Anshika Srivastava
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Taiwanese J. Math. 26(1): 17-32 (February, 2022). DOI: 10.11650/tjm/210702


Let $M$ be a given set of positive integers. A set $S$ of nonnegative integers is said to be an $M$-set if $a,b \in S$ implies $a-b \notin M$. In an unpublished problem collection, Motzkin asked to find maximal upper asymptotic density, denoted by $\mu(M)$, of $M$-sets. The first published work on $\mu(M)$ is due to Cantor and Gordon in 1973, in which, they found the exact value of $\mu(M)$ when $|M| \leq 2$. In fact, this is the only general case, in which, we have a closed formulae for $\mu(M)$. If $|M| \geq 3$, then the exact value of $\mu(M)$ is not known for the general set $M$. In the past six decades or so, several attempts have been given to study $\mu(M)$ but $\mu(M)$ has been found exactly or estimated only in very few cases. In this paper, we study $\mu(M)$ for the families $M = \{ a,a+1,x \}$ and $M = \{ a,a+1,x,y \}$, where $y-x \leq 2$ and $y \gt x \gt a+1$. Our results in the case of $M = \{ a,a+1,x \}$ also give counterexamples to a conjecture of Carraher. Although, different counterexamples to this conjecture, were already given by Liu and Robinson in 2020. We also relate our results with the already know results for the families $M = \{ 1,2,x,x+2 \}$ and $M = \{ 2,3,x,x+2 \}$.


The authors wish to thank to the anonymous referees for providing their valuable comments to improve the paper. We specially thank to the referee during second revision for providing a stronger version of our result in the form of Theorem 3.5 and many other relevant corrections.


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Ram Krishna Pandey. Anshika Srivastava. "Maximal Density of Integral Sets with Missing Differences and the Kappa Values." Taiwanese J. Math. 26 (1) 17 - 32, February, 2022.


Received: 14 November 2020; Revised: 15 March 2021; Accepted: 13 July 2021; Published: February, 2022
First available in Project Euclid: 22 July 2021

MathSciNet: MR4367784
zbMATH: 1484.11031
Digital Object Identifier: 10.11650/tjm/210702

Primary: 11B05 , 11B39

Keywords: $M$-sets , lonely runner conjecture , upper density

Rights: Copyright © 2022 The Mathematical Society of the Republic of China

Vol.26 • No. 1 • February, 2022
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