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OP-11826Status unassessedLogic and foundations

Kakeya sets whose points compute the halting problem — Problem 4.1

Kakeya sets whose points compute the halting problem — Problem 4.1. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today.

MSC 03 · Mathematical logic and foundations

Editorial MSC2020 mapping of Kakeya sets whose points compute the halting problem — Problem 4.1. https://arxiv.org/html/2608.26628v2#S4.Thmproblem1 Not an author-supplied code.

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Problem statement

Problem 4.1.

Is there a Kakeya set A⊆R2A\subseteq\mathbb{R}^{2} such that

x⊕y≥T∅′x\oplus y\geq_{T}\emptyset^{\prime}

for every (x,y)∈A(x,y)\in A?

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Background

  1. Liang Yu

A set A⊆R2A\subseteq\mathbb{R}^{2} is called a Kakeya set if, for every α∈R\alpha\in\mathbb{R}, there exists Cα∈RC_{\alpha}\in\mathbb{R} such that

y=αx+Cα⟹(x,y)∈Ay=\alpha x+C_{\alpha}\quad\Longrightarrow\quad(x,y)\in A

for every (x,y)∈R2(x,y)\in\mathbb{R}^{2}. Equivalently,

{(x,y)∈R2:y=αx+Cα}⊆A\bigl\{(x,y)\in\mathbb{R}^{2}:y=\alpha x+C_{\alpha}\bigr\}\subseteq A

for every α∈R\alpha\in\mathbb{R}.

A Kakeya set AA is called minimal if every proper subset B⊊AB\subsetneq A is not a Kakeya set.

Yinghe Peng has shown that there is a full fine Kakeya set which contains no minimal Kakeya set.

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Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today.

References
  1. [1]Barmpalias, George; Gao, Su; He, Jialiang; Kihara, Takayuki; Nies, Andre; Slaman, Theodore; Tran, Chieu-Minh; Turetsky, Daniel; Welch, Philip; Yu, Liang; Zhang, Hang. Tianyuan Workshop — Open Problems in Mathematical Logic (2026).; Kakeya sets whose points compute the halting problem — Problem 4.1 linkhttp://creativecommons.org/licenses/by/4.0/; attributed adaptation; HTML converted to Markdown with original mathematical notation and source links.

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