Density of sets with missing differences and applications
-
Ram Krishna Pandey
and
Neha Rai
Abstract
For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.
(Communicated by István Gaál)
Acknowledgement
The authors are very much thankful to the anonymous referees for their useful comments to make the paper better.
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© 2021 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Regular papers
- Properties of implication in effect algebras
- On a nonlinear relation for computing the overpartition function
- Six-cycle systems
- Representation of bifinite domains by BF-closure spaces
- L-fuzzy cosets in universal algebras
- Density of sets with missing differences and applications
- Degree of independence of numbers
- A generalization of a result on the sum of element orders of a finite group
- A New fuzzy McShane integrability
- Hankel determinants of second and third order for the class 𝓢 of univalent functions
- Herglotz's theorem for Jacobi-Dunkl positive definite sequences
- Functional inequalities for Gaussian hypergeometric function and generalized elliptic integral of the first kind
- Existence on solutions of a class of casual differential equations on a time scale
- On a general system of difference equations defined by homogeneous functions
- Jordan amenability of banach algebras
- A new notion of orthogonality involving area and length
- Reeb flow invariant ∗-Ricci operators on trans-Sasakian three-manifolds
- A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function
- On first countable quasitopological homotopy groups
