Sequences of Integers with Missing Quotients and Dense Points Without Neighbors

TL;DR

This paper investigates the maximal density of natural number sets avoiding certain rational ratios, linking the problem to lattice point configurations and coloring strategies.

Contribution

It characterizes the maximal asymptotic densities of A-quotient-free sets for specific classes of A, connecting combinatorial and geometric methods.

Findings

Maximal densities are achieved by uniform coloring strategies.

The problem reduces to lattice point counting in geometric regions.

Results extend known cases for coprime integer sets.

Abstract

Let A be a pre-defined set of rational numbers. We say a set of natural numbers S is an A-quotient-free set if no ratio of two elements in S belongs to A. We find the maximal asymptotic density and the maximal upper asymptotic density of A-quotient-free sets when A belongs to a particular class. It is known that in the case A = {p, q}, where p, q are coprime integers greater than one, the latest problem is reduced to evaluation of the largest number of lattice non-adjacent points in a triangle whose legs lie on coordinate axis. We prove that this number is achieved by choosing points of the same color in the checkerboard coloring.