Fractal analysis of Guthrie-Nymann's set and its generalisations

TL;DR

This paper investigates the fractal structure of Guthrie-Nymann's set boundary, showing it can be decomposed into a union of intervals and a zero-measure Cantor set with fractional Hausdorff dimension, extending to a family of such sets.

Contribution

It provides a new fractal analysis of Guthrie-Nymann's set boundary and generalizes the results to a broader family of Cantorvals.

Findings

The boundary can be represented as a union of open intervals and a zero-measure Cantor set.

The Cantor set has a fractional Hausdorff dimension.

Results extend to a countable family of similar Cantorvals.

Abstract

In this paper, we study the fractal properties of the boundary of the Cantorval connected with Guthrie-Nymann's series. In particular, we prove that such a Cantorval can be represented as a union of open intervals and a Cantor set having zero Lebesgue measure and a fractional Hausdorff dimension. Moreover, we extend the result to a countable family of Cantorvals with a similar structure.