Achievement sets -- current results and open problems

TL;DR

The article surveys recent progress and open problems in achievement set theory across one- and higher-dimensional settings, focusing on Kakeya conditions, multigeometric sequences, algebraic sums and decompositions, Cantorvals and their boundaries, planar achievement sets, cardinality functions, and centers of distances. It highlights key structural results such as the Guthrie–Nymann classification, GK25 interior-at-d_C results, and MNP23 decomposition theorems, while foregrounding central open questions (e.g., Jones problems, Cantorval boundary size, planar interior criteria, and center-of-distances realizations). The work emphasizes the interplay between fractal geometry, ergodic/tiling methods, and dynamical-systems perspectives, outlining both substantial partial progress and the many remaining core problems. Overall, it maps a landscape where Kakeya-type criteria offer limited universal tools, while new techniques illuminate the rich topology and measure-theoretic properties of achievement sets and their higher-dimensional analogues.

Abstract

We survey recent developments in the theory of achievement sets and present a substantial collection of open problems.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10