A counterexample to the Karvatskyi--Pratsiovytyi conjecture concerning the achievement set of an intermediate series

TL;DR

This paper presents a counterexample disproving a conjecture about the topological nature of achievement sets of intermediate series, and proposes an improved conjecture that remains unrefuted.

Contribution

The authors provide the first counterexample to the Karvatskyi--Pratsiovytyi conjecture and suggest an improved version of the conjecture.

Findings

Counterexample invalidates the original conjecture.

Proposed an improved conjecture not refuted by the counterexample.

Clarified the topological properties of achievement sets.

Abstract

We found a counterexample to the conjecture of Karvatskyi and Pratsiovytyi concerning the topological type of the achievement set of an intermediate series (Proceedings of the International Geometry Center, 2023. https://doi.org/10.15673/pigc.v16i3.2519). This conjecture is based on an analogy with the squeeze theorem from calculus. We also proposed an improved version of the conjecture, which this counterexample does not refute.