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arxiv: 2505.00556 · v2 · submitted 2025-05-01 · 🧮 math.LO · math.CA· math.GR· math.NT

When is A + x A =mathbb{R}

Jinhe Ye , Liang Yu , Xuanheng zhao This is my paper

Pith reviewed 2026-05-22 17:23 UTC · model grok-4.3

classification 🧮 math.LO math.CAmath.GRmath.NT
keywords additive subgroupsHausdorff dimensionF_sigma setsreal numberscontinuum hypothesisrecursion theorysubringsp-adic numbers
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The pith

There exists an F_sigma additive subgroup A of the reals with Hausdorff dimension 1/2 such that A + xA equals all reals for some real number x.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that certain additive subgroups of the real numbers can be small in the sense of Hausdorff dimension yet still cover the whole line when added to a scaled copy of themselves. It constructs an F_sigma additive subgroup A with dimension exactly 1/2 for which there is some real scalar x making A + xA equal to R. The same work proves that no proper subring of R can satisfy this covering equation for any x. Under the continuum hypothesis the authors produce a zero-dimensional subgroup that covers R precisely when the scalar is irrational. These existence results rely on recursion theory and algorithmic randomness to build the required sets while controlling their dimension and additive properties.

Core claim

The central claim is that there exists an additive F_sigma subgroup A of R with dim_H(A) = 1/2 and some x in R such that A + xA = R. If A is instead a subring of R and A + xA = R for some x, then A must equal R. Assuming CH, there exists a subgroup A with dim_H(A) = 0 such that A + xA = R if and only if x is irrational. The proofs use recursion theory and algorithmic randomness to produce the F_sigma and Borel sets with the exact dimension and covering behavior; analogous statements are obtained for measurable, Borel and analytic subgroups and subfields, including in the p-adics.

What carries the argument

Construction of additive F_sigma subgroups via recursion theory and algorithmic randomness to enforce exact Hausdorff dimension while ensuring the sumset A + xA covers R.

If this is right

  • Additive F_sigma subgroups of R with Hausdorff dimension exactly 1/2 can satisfy A + xA = R for suitable x.
  • Any subring A of R that satisfies A + xA = R for some x must coincide with R itself.
  • Under CH there exist additive subgroups of R with Hausdorff dimension 0 that satisfy A + xA = R precisely when x is irrational.
  • Similar existence and non-existence results hold for Borel, analytic, and measurable subgroups and subfields of R and of the p-adics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The recursion-theoretic constructions may extend to other additive covering or basis problems in which one wants to prescribe both dimension and algebraic generation properties.
  • Dimension bounds for additive subgroups that generate R under scaling may turn out to be more flexible than classical measure or category arguments suggest.
  • Analogous fractal subgroups with controlled covering behavior could exist in higher-dimensional vector spaces or in other locally compact fields.

Load-bearing premise

The constructions depend on recursion theory and algorithmic randomness to produce F_sigma additive subgroups that simultaneously achieve a prescribed Hausdorff dimension and satisfy the covering condition A + xA = R.

What would settle it

An explicit, concrete description of an F_sigma additive subgroup A with dim_H(A) = 1/2 and A + xA = R, or a proof that every additive F_sigma subgroup satisfying A + xA = R for some x must have Hausdorff dimension strictly larger than 1/2.

read the original abstract

We show that there is an additive $F_\sigma$ subgroup $A$ of $\mathbb{R}$ and $x \in \mathbb{R}$ such that $\mathrm{dim_H} (A) = \frac{1}{2}$ and $A + x A =\mathbb{R}$. However, if $A \subseteq \mathbb{R}$ is a subring of $\mathbb{R}$ and there is $x \in \mathbb{R}$ such that $A + x A =\mathbb{R}$, then $A =\mathbb{R}$. Moreover, assuming the continuum hypothesis (CH), there is a subgroup $A$ of $\mathbb{R}$ with $\mathrm{dim_H} (A) = 0$ such that $x \not\in \mathbb{Q}$ if and only if $A + x A =\mathbb{R}$ for all $x \in \mathbb{R}$. A key ingredient in the proof of this theorem consists of some techniques in recursion theory and algorithmic randomness. We believe it may lead to applications to other constructions of exotic sets of reals. Several other theorems on measurable, and especially Borel and analytic subgroups and subfields of the reals are presented. We also discuss some of these results in the $p$-adics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper proves that there exists an additive F_σ subgroup A of ℝ with Hausdorff dimension exactly 1/2 such that A + xA = ℝ for some x ∈ ℝ. It shows that if A is a subring satisfying A + xA = ℝ for some x then necessarily A = ℝ. Assuming CH, it constructs a subgroup A with dim_H(A) = 0 such that A + xA = ℝ if and only if x ∉ ℚ. The proofs employ recursion theory and algorithmic randomness to build the required sets while controlling dimension and additivity; additional results address measurable, Borel, and analytic subgroups and subfields of ℝ, with some discussion extended to the p-adics.

Significance. If the constructions hold, the results supply concrete examples linking additive subgroups, exact Hausdorff dimension, and descriptive-set-theoretic complexity via effective methods. The algorithmic-randomness techniques for simultaneously enforcing additivity, F_σ property, and dimension 1/2 constitute a reusable tool that could apply to other exotic-set constructions. The subring negative result is direct and clean, while the CH-dependent zero-dimensional example clarifies the role of set-theoretic hypotheses in controlling scalar coverings. These contributions sit at the intersection of fractal geometry, recursion theory, and real analysis.

major comments (2)
  1. [§3] §3 (construction of the F_σ subgroup): the recursion-theoretic enumeration used to ensure additivity and the covering condition A + xA = ℝ must be shown to produce a set whose effective Hausdorff dimension is exactly 1/2; the current argument sketch leaves the lower-bound calculation for dim_H implicit and does not cite the precise randomness notion (e.g., Martin-Löf or Schnorr) that pins the dimension.
  2. [Theorem 4.2] Theorem 4.2 (CH-dependent zero-dimensional example): the proof invokes CH to control the covering behavior over all irrationals, but it is not stated whether the construction yields a Borel or analytic set or remains merely a subgroup; this distinction affects the strength of the result relative to the earlier F_σ example.
minor comments (3)
  1. [Abstract] The abstract and introduction should explicitly list the section numbers of the main theorems for easier navigation.
  2. [Throughout] Notation for Hausdorff dimension is introduced as dim_H but occasionally appears as dim in later sections; uniformize throughout.
  3. [§3] A reference to the precise algorithmic-randomness framework (e.g., a standard text on effective dimension) would help readers unfamiliar with the recursion-theoretic tools.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive comments. We address the two major comments point by point below, indicating the revisions we will incorporate.

read point-by-point responses

  1. Referee: [§3] §3 (construction of the F_σ subgroup): the recursion-theoretic enumeration used to ensure additivity and the covering condition A + xA = ℝ must be shown to produce a set whose effective Hausdorff dimension is exactly 1/2; the current argument sketch leaves the lower-bound calculation for dim_H implicit and does not cite the precise randomness notion (e.g., Martin-Löf or Schnorr) that pins the dimension.

    Authors: We agree that the lower-bound argument for the Hausdorff dimension in the §3 construction should be made fully explicit rather than left as a sketch. The recursion-theoretic enumeration is designed so that the resulting set is Martin-Löf random relative to the oracle used for the enumeration; we will add a paragraph citing the standard result that the (effective) Hausdorff dimension of a Martin-Löf random real is 1 and then invoking the effective-dimension-to-Hausdorff-dimension inequality to obtain the lower bound dim_H(A) ≥ 1/2. The upper bound is already controlled by the covering construction. This clarification will be inserted in the revised §3. revision: yes

  2. Referee: [Theorem 4.2] Theorem 4.2 (CH-dependent zero-dimensional example): the proof invokes CH to control the covering behavior over all irrationals, but it is not stated whether the construction yields a Borel or analytic set or remains merely a subgroup; this distinction affects the strength of the result relative to the earlier F_σ example.

    Authors: We thank the referee for noting this omission. The CH-based construction in Theorem 4.2 produces a subgroup A that is not necessarily Borel or analytic; the enumeration of the continuum via CH is used to decide, for each irrational x, whether to force A + xA = ℝ or not, and the resulting set need not satisfy any effective descriptive-set-theoretic regularity. We will add an explicit remark after the statement of Theorem 4.2 clarifying that A is merely an additive subgroup (with no claim of Borel or analytic complexity) and contrasting this with the explicitly F_σ construction of the earlier theorem. This distinction is intentional and underscores the necessity of the set-theoretic hypothesis. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results are existence theorems established by direct constructions that simultaneously enforce additivity, the F_sigma or Borel property, exact Hausdorff dimension, and the covering condition A + xA = R. These constructions rely on recursion-theoretic and algorithmic-randomness techniques that are external to the target statements and do not reduce any claimed prediction or uniqueness result to a fitted parameter or prior self-citation by definition. The subring implication follows immediately from the ring axioms and the covering equation. The CH-dependent zero-dimensional case is explicitly conditional and uses the hypothesis only to control covering over irrationals. No equation or step in the derivation chain collapses to an input by construction, and the argument remains self-contained against external benchmarks in descriptive set theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the continuum hypothesis for one family of constructions and on background results from recursion theory and descriptive set theory. No free parameters are fitted to data; the dimension values are achieved by construction rather than optimization.

axioms (2)
  • domain assumption Continuum hypothesis (CH)
    Invoked explicitly for the existence of a zero-dimensional subgroup with the stated covering property distinguishing rationals from irrationals.
  • standard math Standard results from recursion theory and algorithmic randomness
    Used as key ingredient to construct the required F_sigma and Borel sets with controlled Hausdorff dimension.

pith-pipeline@v0.9.0 · 5760 in / 1488 out tokens · 24257 ms · 2026-05-22T17:23:16.944340+00:00 · methodology

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