Open Problems in Mathematical Logic — Problem 11

Definitions and mathematical statement

For sets $B,D\subseteq{\mathbb{N}}$, their Turing join is \[ B\oplus D=\{2n:n\in B\}\cup\{2n+1:n\in D\}. \] Write $B\leq_T D$ when membership in $B$ is computable by a Turing machine with oracle $D$. Let $\emptyset'$ denote the halting set for a fixed effective enumeration of Turing machines.

The source applies Turing join and reducibility to real numbers without specifying how real numbers are represented as oracles. To express the question precisely relative to a representation, let $c:{\mathbb{R}}\to\mathcal P({\mathbb{N}})$ denote a fixed coding convention for real numbers. The choice of $c$ is an unresolved source qualification; it is not an additional universally quantified parameter in the question.

Following the source's definition, a set $A\subseteq{\mathbb{R}}^2$ is a Kakeya set if \[ \forall\alpha\in{\mathbb{R}}\;\exists C\in{\mathbb{R}}\;\forall t\in{\mathbb{R}}, \qquad (t,\alpha t+C)\in A. \] Thus the required objects are entire affine lines of every finite real slope. The definition does not require a vertical line, and it imposes no measurability or other regularity condition on $A$.

Relative to the intended, but unspecified, real-oracle coding $c$, the question is whether \[ \exists A\subseteq{\mathbb{R}}^2\quad \left[ \begin{aligned} &\forall\alpha\in{\mathbb{R}}\;\exists C\in{\mathbb{R}}\;\forall t\in{\mathbb{R}}, && (t,\alpha t+C)\in A,\\ &\forall(x,y)\in A, && \emptyset'\leq_T c(x)\oplus c(y) \end{aligned} \right]. \] The source writes the second condition as $x\oplus y\geq_T\emptyset'$. Identifying the intended representation of real oracles remains necessary to resolve that notation completely.

Short English statement

Does there exist a subset of ${\mathbb{R}}^2$ containing a whole line of every finite real slope such that the two coordinates of every point jointly compute the halting set? The source does not specify the real-oracle representation used in this computability condition.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800999$, exact source problem code catalogue entry, title "Open Problems in Mathematical Logic --- Problem 11." The record attributes the entry to Barmpalias, Gao et al., with proposed year 2026, and labels its status as open; these are supplied metadata, not independently verified conclusions.
• Supplied source reference: Barmpalias, Gao et al., Open Problems in Mathematical Logic (2026), Problem 11, https://arxiv.org/abs/2608.26628. The background records access on 2026-10-16 and extraction by arxiv_latex. These provenance details are supplied by the record; the reference and its contents have not been independently checked for this draft.