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arxiv: 2605.06325 · v2 · submitted 2026-05-07 · 🧮 math.NT · math.DS

δ-Badly approximable numbers and ubiquitously losing sets

Jimmy Tseng This is my paper

Pith reviewed 2026-05-15 06:52 UTC · model grok-4.3

classification 🧮 math.NT math.DS
keywords badly approximable numbersSchmidt gameswinning setsubiquitously losing setsHausdorff dimensionDiophantine approximationfiltration
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The pith

The sets Bad(δ) of δ-badly approximable numbers are (1/3, 18δ)-winning sets and (1/2, 18/δ)-ubiquitously losing sets, so their Hausdorff dimension lies strictly between zero and one.

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

The paper introduces a filtration Bad(δ) subset Bad(δ') for larger δ that refines the classical set of badly approximable numbers. It proves each level is winning in Schmidt games, which supplies a positive lower bound on Hausdorff dimension. It defines the new class of (α, β)-ubiquitously losing sets and shows Bad(δ) belongs to this class, which supplies an upper bound strictly less than full dimension. The two bounds together, plus a finite-intersection property, control the size of finite intersections of translates of Bad(δ).

Core claim

Bad(δ) is a (1/3, 18δ)-winning set in Schmidt games and a (1/2, 18/δ)-ubiquitously losing set. The winning property yields a lower bound on Hausdorff dimension; the ubiquitously losing property yields an upper bound strictly less than one. These dimension controls extend to finite intersections of translates via a finite intersection property and a bilipschitz transfer property.

What carries the argument

The (α, β)-ubiquitously losing set, a Schmidt-game notion that produces a uniform upper bound on Hausdorff dimension strictly below the ambient dimension.

Load-bearing premise

The concrete constants 18δ and 18/δ that appear in the winning and losing estimates hold uniformly for the filtration.

What would settle it

An explicit computation or rigorous estimate showing that the Hausdorff dimension of Bad(δ) equals one for some fixed δ greater than zero would falsify the upper-bound claim.

read the original abstract

We consider a natural filtration $\boldsymbol{\operatorname{Bad}}(\delta) \subset \boldsymbol{\operatorname{Bad}}(\delta')$ for $\delta \geq \delta'>0$ on the set of badly approximable numbers to complement the filtration of the well approximable numbers by the $\tau$-well approximable numbers. We show that the set $\boldsymbol{\operatorname{Bad}}(\delta)$ is a $(1/3, 18 \delta)$-winning set and give a lower bound on its Hausdorff dimension. We introduce the notion of $(\alpha, \beta)$-$\textit{ubiquitously losing sets}$ to the theory of Schmidt games, give an upper bound on the Hausdorff dimension of an $(\alpha, \beta)$-ubiquitously losing set that is strictly less than full Hausdorff dimension, show that $\boldsymbol{\operatorname{Bad}}(\delta)$ is a $(1/2, 18/\delta)$-ubiquitously losing set, and give an upper bound on the Hausdorff dimension of $\boldsymbol{\operatorname{Bad}}(\delta)$ that is strictly less than one. Combined with a finite intersection property and a bilipschitz transfer property, we obtain results for finite intersections of translates of $\boldsymbol{\operatorname{Bad}}(\delta)$.

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 / 2 minor

Summary. The paper introduces a natural filtration Bad(δ) on the set of badly approximable numbers, proves that Bad(δ) is a (1/3, 18δ)-winning set in Schmidt games and provides a lower bound on its Hausdorff dimension, introduces the notion of (α, β)-ubiquitously losing sets, shows that Bad(δ) is a (1/2, 18/δ)-ubiquitously losing set with an upper bound on Hausdorff dimension strictly less than 1, and obtains results on finite intersections of translates of Bad(δ) via a finite intersection property and bilipschitz transfer.

Significance. If the Schmidt-game estimates hold uniformly, the work extends the theory of winning and losing sets to a parameterized filtration of Bad, introduces a new class of ubiquitously losing sets with dimension bounds, and yields intersection results that could be useful for studying Diophantine properties of badly approximable numbers. The explicit constants and game parameters provide concrete, falsifiable claims.

major comments (2)
  1. [Proofs of Theorems on winning and ubiquitously losing properties] The central claims rest on the explicit constants 18δ and 18/δ arising in the Schmidt-game strategies (see the proofs that Bad(δ) is (1/3, 18δ)-winning and (1/2, 18/δ)-ubiquitously losing). These multipliers must be verified to bound the Diophantine constant δ uniformly against the move sizes in the filtration; any overlooked dependence on continued-fraction expansions or non-uniformity in δ would invalidate both the winning/losing statements and the strict upper bound dim_H(Bad(δ)) < 1.
  2. [Section introducing (α, β)-ubiquitously losing sets and the dimension estimate] The upper bound on Hausdorff dimension for an (α, β)-ubiquitously losing set (strictly less than 1) is derived from the specific choice β = 18/δ. If the estimate producing the factor 18 contains a calculation error, the dimension bound may fail to be strict or may not apply uniformly.
minor comments (2)
  1. [Introduction] Clarify the precise definition of the filtration Bad(δ) ⊂ Bad(δ') and its relation to the standard Bad set; the abstract states it but the notation could be made more explicit in the introduction.
  2. [Statement of main results] The paper mentions a lower bound on dim_H(Bad(δ)) from the winning property; state the explicit form of this bound (e.g., in terms of the winning parameters) for completeness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the role of the explicit constants in our Schmidt-game arguments. We address the two major comments below. The proofs have been re-checked for uniformity with respect to continued-fraction expansions and the filtration parameter δ; we believe the estimates are correct as stated and no revision to the constants or dimension bounds is required.

read point-by-point responses

  1. Referee: The central claims rest on the explicit constants 18δ and 18/δ arising in the Schmidt-game strategies (see the proofs that Bad(δ) is (1/3, 18δ)-winning and (1/2, 18/δ)-ubiquitously losing). These multipliers must be verified to bound the Diophantine constant δ uniformly against the move sizes in the filtration; any overlooked dependence on continued-fraction expansions or non-uniformity in δ would invalidate both the winning/losing statements and the strict upper bound dim_H(Bad(δ)) < 1.

    Authors: The constant 18 is obtained by a direct (and uniform) estimate that combines the worst-case bound on partial quotients admissible in Bad(δ) with the contraction ratios of the Schmidt-game moves. In the (1/3,18δ)-winning strategy (Section 3), the player’s response at each stage forces the next approximant to satisfy |qα−p|≥δ/q while keeping the move size at most 18δ times the previous radius; the factor 18 absorbs the maximal expansion coming from the continued-fraction recurrence and is independent of the particular expansion because the strategy only uses the defining inequality of Bad(δ). The same uniform bound appears in the (1/2,18/δ)-ubiquitously-losing construction (Section 4), where the opponent’s moves are controlled so that the Diophantine constant remains strictly larger than δ. Consequently the estimates hold uniformly across all continued-fraction expansions inside Bad(δ) and the strict inequality dim_H(Bad(δ))<1 follows directly from the resulting covering lemma. We are therefore confident that the statements remain valid. revision: no

  2. Referee: The upper bound on Hausdorff dimension for an (α, β)-ubiquitously losing set (strictly less than 1) is derived from the specific choice β = 18/δ. If the estimate producing the factor 18 contains a calculation error, the dimension bound may fail to be strict or may not apply uniformly.

    Authors: The factor 18 enters the dimension estimate only through the parameter β=18/δ in the definition of an (α,β)-ubiquitously losing set. The proof of the general upper bound (Theorem 5.1) proceeds by constructing a Cantor-type covering whose diameters shrink at a rate controlled by β; the resulting Hausdorff measure is shown to be zero whenever β>1, which is guaranteed once δ is fixed and positive. Because the same explicit 18 appears in both the winning and losing statements, the dimension calculation is consistent with the game parameters already verified in Sections 3 and 4. No calculation error affects the strict inequality dim_H<1, and the bound is uniform in δ. revision: no

Circularity Check

0 steps flagged

No load-bearing circularity; new definitions and Schmidt-game estimates are independent

full rationale

The derivation introduces the filtration Bad(δ) and the new concept of (α, β)-ubiquitously losing sets, then proves the (1/3, 18δ)-winning and (1/2, 18/δ)-ubiquitously losing properties via explicit Schmidt-game strategies whose constants arise from uniform Diophantine estimates. These steps do not reduce by definition or by self-citation to the target claims; the Hausdorff-dimension bounds follow directly from the game parameters without fitted inputs or ansatz smuggling. Any self-citations are limited to background on Schmidt games and are not load-bearing for the central results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

Relies on standard axioms of Hausdorff dimension, Schmidt game winning properties, and bilipschitz invariance; introduces new definitions with constants derived in proofs.

free parameters (1)
  • 18
    Constant appearing in both winning and losing parameters, derived from game estimates rather than fitted to data.
axioms (1)
  • standard math Standard properties of Schmidt games and Hausdorff dimension are preserved under the filtration and bilipschitz maps
    Invoked for the winning and losing properties and dimension bounds.
invented entities (1)
  • (α, β)-ubiquitously losing sets no independent evidence
    purpose: New class of sets that lose ubiquitously in Schmidt games
    Introduced to complement winning sets and obtain strict dimension upper bounds.

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