Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}
-
Ram Krishna Pandey
and
Neha Rai
ABSTRACT
Let M be a set of positive integers. We study the maximal density μ(M) of the sets of nonnegative integers S whose elements do not differ by an element in M. In 1973, Cantor and Gordon established a formula for μ(M) for |M| ≤ 2. Since then, many researchers have worked upon the problem and found several partial results in the case |M| ≥ 3, including some results in the case when M is an infinite set. In this paper, we study the maximal density problem for the families M = {a, a+1, 2a+1, n} and M = {a, a+1, 2a+1, 3a+1, n}, where a and n are positive integers and n is sufficiently large. In most of the cases, we find bounds for the parameter kappa, denoted by κ(M), which actually serves as a lower bound for μ(M). The parameter κ(M) has already got its importance due to its rich connection with problems such as the “lonely runner conjecture” in Diophantine approximation and coloring parameters such as “circular coloring” and “fractional coloring” in graph theory. We also give some partial results for the general family M = {a, a +1, 2a + 1,…, (s – 2)a + 1, n}, where s ≥ 5 and mention related problems in the remaining cases for future work.
Funding statement: The first author is thankful to the Council of Scientific and Industrial Research (CSIR) for providing the research grant no. 25(0314)/20/EMR-II.
Acknowledgement
The authors thank the referees for their very useful suggestions which considerably improved the paper.
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Articles in the same Issue
- Parameters in Inversion Sequences
- On the Pecking Order Between Those of Mitsch and Clifford
- A Note on Cuts of Lattice-Valued Functions and Concept Lattices
- Trigonometric Sums and Riemann Zeta Function
- Notes on the Equation d(n) = d(φ(n)) and Related Inequalities
- Harmony of Asymmetric Variants of the Filbert and Lilbert Matrices in q-form
- Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}
- Extensions in Time Scales Integral Inequalities of Jensen’s Type via Fink’s Identity
- A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions
- Extended Bromwich-Hansen Series
- Iterative Criteria for Oscillation of Third-Order Delay Differential Equations with p-Laplacian Operator
- Vallée-Poussin Theorem for Equations with Caputo Fractional Derivative
- On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations
- On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces
- Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation
- On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers
- The Smallest and the Largest Families of Some Classes of 𝒜-Continuous Functions
- Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications
- Strong Consistency of Least-Squares Estimators in the Simple Linear Errors-in-Variables Regression Model with Widely Orthant Dependent Random Variables
