From Erdos Problem 1154 to a Zero One Law for Turing Ideals

Abstract: Erdős Problem 1154 asks whether every number in $[0,1]$ occurs as the Hausdorff dimension of a subring or subfield of $\mathbb{R}$. Motivated by this problem, Liang Yu asked whether the reals of an inner model can have Hausdorff dimension strictly between zero and one when their dimension is computed in an outer model. We prove a stronger result: if $\mathcal I \subseteq 2<sup>ω$ is any Turing ideal, then $\dim_{\mathrm H} \mathcal I \in {0,1}$. Equivalently, the real-closed field whose reals have Turing degrees in $\mathcal I$ has Hausdorff dimension zero or one. The proof combines digit interleaving with the Furstenberg-set theorem of Orponen and Shmerkin.

• The paper proves that every Turing ideal, viewed as its associated real-closed field, has Hausdorff dimension exactly 0 or 1, ruling out intermediate dimensions without definability, measurability, or absoluteness assumptions.
• The proof transfers dimension through binary evaluation, uses Turing-join closure to double product dimensions, and derives a contradiction from the Orponen–Shmerkin Furstenberg-set theorem whenever 0 < d < 1.
• The result applies to reals of any transitive inner model inside a larger ZFC model, answering Liang Yu’s question negatively while leaving the broader Erdős Problem 1154 for arbitrary subfields open.

This paper resolves a set-theoretic variant of a classical Erdős problem by proving that the Hausdorff dimension of any Turing ideal, computed via its associated real-closed field, is either zero or one (2608.18955). The result answers negatively a question of Liang Yu concerning the dimension of the reals of inner models, and does so without any definability, measurability, or absoluteness assumptions beyond transitivity.

Background and problem

Erdős and Volkmann showed that every $\alpha \in [0,1]$ occurs as the Hausdorff dimension of an additive subgroup of $\mathbb{R}$, but closure under multiplication imposes rigidity: Falconer proved that Borel or analytic subrings cannot have dimension strictly between $1/2$ and $1$, and Edgar–Miller established the full zero–one law for analytic subrings. On the other hand, Mauldin constructed subfields of every prescribed dimension under CH. Erdős Problem 1154 asks whether every $\alpha$ arises as the dimension of some subring or subfield of $\mathbb{R}$. The paper addresses the sharpened question posed by Yu: if $M \subseteq N$ are models of ZFC with $M$ transitive and definable in $N$, can $N$ compute $\mathbb{R}$0 to be strictly between zero and one? In particular, what happens for $\mathbb{R}$1?

The answer is no, uniformly over all Turing ideals. For a Turing ideal $\mathbb{R}$2, define $\mathbb{R}$3, equivalently the directed union $\mathbb{R}$4 where $\mathbb{R}$5 denotes the field of $\mathbb{R}$6-computable reals.

Structure of the proof

The argument proceeds in four steps, each isolated as a lemma.

Real-closedness. Since $\mathbb{R}$7 is directed under the Turing join and each $\mathbb{R}$8 is real closed, $\mathbb{R}$9 is real closed: the finitely many coefficients of any polynomial lie in a single member of the directed family, along with its real algebraic roots.

Dimension transfer via binary evaluation. The map $1/2$0 from Cantor space to $1/2$1 preserves Hausdorff dimension (Lipschitz, with inverse images of intervals covered by boundedly many cylinders), and the ambiguity of binary expansions is countable and hence negligible. Downward closure of Turing ideals gives $1/2$2, and since $1/2$3 contains $1/2$4, it is a countable union of translates of this intersection. Countable stability then yields $1/2$5, and similarly for products.

Doubling under the join. The join map $1/2$6 is a bijection from $1/2$7 onto $1/2$8, bi-Lipschitz when the domain carries the squared maximum product metric. Because squaring a metric halves Hausdorff dimension, if $1/2$9 then $1$0. This is the combinatorial core, exploiting exactly the join-closure axiom defining a Turing ideal.

Contradiction via Furstenberg sets. Assuming $1$1, the family of lines with slope and intercept in $1$2 has parameter-set dimension $1$3, and each such line meets $1$4 in a copy of $1$5, of dimension $1$6. Applying the Orponen–Shmerkin Furstenberg-set theorem — which requires no measurability hypothesis — there exists $1$7 with

$1$8

contradicting the previous step. Hence $1$9.

Inner-model consequence

Taking $\alpha$0 inside a transitive model $\alpha$1: transitivity plus closure of $\alpha$2 under recursive constructions make $\alpha$3 a Turing ideal in $\alpha$4, binary evaluation is absolute between $\alpha$5 and $\alpha$6, so $\alpha$7. The main theorem, computed in $\alpha$8, yields $\alpha$9 for all transitive pairs $\mathbb{R}$0 of ZFC models. Notably, the proof uses neither definability of $\mathbb{R}$1 in $\mathbb{R}$2 nor equality of ordinals, so it applies well beyond the $\mathbb{R}$3 case Yu singled out.

Limitations and open questions

The paper's scope is deliberately restricted to sets arising as Turing ideals; it does not settle Erdős Problem 1154 itself, which concerns arbitrary subfields and remains open in ZFC — Mauldin's CH construction shows the general statement is independent-flavored. The reduction to the Orponen–Shmerkin theorem means the quantitative content depends on their $\mathbb{R}$4, which is not explicit here. Two natural questions remain open within this framework: whether the zero value is realized precisely by countable-generated ideals (i.e., ideals generated under joins by a single real), and whether analogous dichotomies hold for other degree structures, such as enumeration-degree ideals or hyperarithmetic analogues, where the corresponding Furstenberg-type estimate would need replacement.

Conclusion

The paper establishes that every Turing ideal, hence every class of reals closed downward under Turing reducibility and under finite joins, has Hausdorff dimension either zero or one, answering Yu's inner-model question negatively and uniformly. Methodologically, it connects effective descriptive set theory with contemporary additive combinatorics, using join-closure to force a dimensional doubling that contradicts Furstenberg-set lower bounds. The argument is short, assumption-free on the ideal side, and transfers verbatim to transitive inner models.