A note on sets avoiding rational distances in category bases

TL;DR

This paper generalizes existing results on subsets avoiding rational distances within sets in Euclidean spaces to certain types of category bases, expanding the theoretical understanding of distance properties in these mathematical structures.

Contribution

It extends theorems about rational distance avoidance from Euclidean spaces to specific category bases, broadening the scope of these geometric measure results.

Findings

Existence of rational distance avoiding subsets in certain category bases

Generalization of Kumar's theorem to new mathematical structures

Insights into the structure of sets with no rational distances

Abstract

Michalski gave a short and elegant proof of a theorem of A. Kumar which states that for each set A in R, there exists a subset B of A which is full in A and such that no distance between points in B is a rational number. He also proved a similar theorem for sets in the Euclidean plane. In this paper, we generalize these results in some special types of category bases.