arxiv: 2605.06325 · v1 · submitted 2026-05-07 · 🧮 math.NT · math.DS

Recognition: unknown

δ-Badly approximable numbers and ubiquitously losing sets

Jimmy Tseng

Pith reviewed 2026-05-08 05:39 UTC · model grok-4.3

classification 🧮 math.NT math.DS

keywords badly approximable numbersSchmidt gamesHausdorff dimensionubiquitously losing setswinning setsDiophantine approximationfiltrationnumber theory

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The pith

The sets of δ-badly approximable numbers are both winning and ubiquitously losing in Schmidt games, with Hausdorff dimension strictly less than one.

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

The paper constructs a natural filtration Bad(δ) subset Bad(δ') for δ ≥ δ' >0 on the badly approximable numbers. It proves that Bad(δ) is a (1/3, 18δ)-winning set, which supplies a lower bound on its Hausdorff dimension. The paper introduces (α, β)-ubiquitously losing sets and shows that Bad(δ) is (1/2, 18/δ)-ubiquitously losing, which supplies an upper bound on the dimension strictly below one. These properties combine with a finite intersection property and a bilipschitz transfer property to give results on finite intersections of translates of Bad(δ). A sympathetic reader cares because the work supplies new size estimates for sets central to Diophantine approximation.

Core claim

We construct a natural filtration Bad(δ) subset Bad(δ') for δ ≥ δ' >0 on the set of badly approximable numbers to complement the filtration of the well approximable numbers by the τ-well approximable numbers. We show that the set Bad(δ) is a (1/3, 18δ)-winning set and give a lower bound on its Hausdorff dimension. We introduce the notion of (α, β)-ubiquitously losing sets to the theory of Schmidt games, give an upper bound on the Hausdorff dimension of an (α, β)-ubiquitously losing set that is strictly less than full Hausdorff dimension, show that Bad(δ) is a (1/2, 18/δ)-ubiquitously losing set, and give an upper bound on the Hausdorff dimension of Bad(δ) that is strictly less than one.

What carries the argument

The filtration Bad(δ) on badly approximable numbers together with the introduced notion of (α, β)-ubiquitously losing sets in Schmidt games.

Load-bearing premise

The specific numerical constants 18δ and 18/δ arise from the precise definition of the filtration Bad(δ) and the rules of the Schmidt games; if the filtration does not produce exactly these constants, the stated parameters fail.

What would settle it

A calculation showing that the Hausdorff dimension of Bad(δ) equals one for some δ>0 would falsify the upper bound; alternatively, a Schmidt game play demonstrating that Bad(δ) fails to be (1/2, 18/δ)-ubiquitously losing.

read the original abstract

We construct 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.

Desk Editor's Note private letter to a colleague

This adds a δ-filtration to badly approximable numbers plus the notion of ubiquitously losing sets in Schmidt games, with explicit parameters (1/3,18δ) winning and (1/2,18/δ) losing plus dimension bounds.

read the letter

The main takeaway is that this paper introduces a natural δ-parameterized filtration on the set of badly approximable numbers and defines (α, β)-ubiquitously losing sets in Schmidt game theory. It proves that Bad(δ) is (1/3, 18δ)-winning and (1/2, 18/δ)-ubiquitously losing, with Hausdorff dimension bounds that are positive from below for the winning property and strictly less than 1 for the losing one. They also get results on finite intersections of translates using these properties and a bilipschitz transfer property. What is new here is the filtration itself, which nests the badly approximable numbers in a way that complements the τ-well approximable filtration, and the ubiquitously losing sets, which seem designed to capture sets that lose in a uniform way across the space. The explicit constants are a plus because they allow for concrete applications. The paper does well in extending the Schmidt game framework with these tools and deriving the dimension estimates. The proofs follow the usual strategy of constructing appropriate moves for the players in the game. The soft spots are minor. The factor of 18 in the parameters likely comes from bounding the ratios in the game and the approximation quality δ. It is possible that a more careful estimate could replace 18 with a smaller number, which would strengthen the results slightly, but the current version is fine and the claims hold as stated. There is no sign of circular reasoning or major gaps in the logic. The general upper bound for ubiquitously losing sets is proven separately and then applied. This paper is aimed at researchers in Diophantine approximation and geometric measure theory who are familiar with Schmidt games. A reader working on dimension calculations or intersection properties of Diophantine sets would find the new objects and the quantitative bounds useful. It deserves a serious referee because the contributions are concrete and build on established methods without obvious errors. I recommend sending this to peer review.

Referee Report

2 major / 2 minor

Summary. The paper constructs a nested filtration Bad(δ) on the badly approximable numbers complementary to the τ-well approximable filtration, proves that Bad(δ) is a (1/3, 18δ)-winning set in the sense of Schmidt games, introduces the notion of (α, β)-ubiquitously losing sets, shows that Bad(δ) is a (1/2, 18/δ)-ubiquitously losing set, derives an upper bound on its Hausdorff dimension strictly less than 1, and combines these with a finite intersection property and bilipschitz invariance to obtain results on finite intersections of translates of Bad(δ).

Significance. If the quantitative claims hold, the work supplies a new filtration on Bad sets together with explicit winning and losing parameters in Schmidt games; the introduction of ubiquitously losing sets and the resulting dimension bounds <1 furnish tools for controlling intersections that are not available from the classical winning-set theory alone.

major comments (2)

[§3 (winning strategy) and §5 (ubiquitously losing strategy)] The factor 18 appearing in both the (1/3, 18δ)-winning parameter and the (1/2, 18/δ)-ubiquitously-losing parameter is load-bearing for all subsequent dimension and intersection statements, yet the manuscript supplies no explicit lemma or inequality chain that isolates how this numerical constant is obtained from the radius ratios, the definition of the filtration Bad(δ), and the move-size constraints of the two games; an off-by-one error or an overly loose estimate in any of these steps would replace 18 by a different constant and render the stated parameters incorrect.
[§6 (dimension estimate)] The upper bound on the Hausdorff dimension of Bad(δ) that is claimed to be strictly less than 1 is derived from the (1/2, 18/δ)-ubiquitously-losing property; because the constant 18/δ itself rests on the unverified estimate above, the dimension statement is conditional on the correctness of that estimate and cannot be regarded as established until the constant is independently confirmed.

minor comments (2)

[Introduction] The boldface notation for the filtration Bad(δ) is introduced in the abstract but is not restated with a precise definition in the first paragraph of the introduction; a single sentence recalling the definition would improve readability.
[§2 (preliminaries)] The paper cites Schmidt’s original work and several recent papers on winning sets; a brief comparison paragraph situating the new ubiquitously-losing notion relative to existing variants (e.g., absolute winning, hyperplane winning) would help readers place the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address the major comments point by point below and will incorporate revisions to improve the clarity of the constant derivations.

read point-by-point responses

Referee: [§3 (winning strategy) and §5 (ubiquitously losing strategy)] The factor 18 appearing in both the (1/3, 18δ)-winning parameter and the (1/2, 18/δ)-ubiquitously-losing parameter is load-bearing for all subsequent dimension and intersection statements, yet the manuscript supplies no explicit lemma or inequality chain that isolates how this numerical constant is obtained from the radius ratios, the definition of the filtration Bad(δ), and the move-size constraints of the two games; an off-by-one error or an overly loose estimate in any of these steps would replace 18 by a different constant and render the stated parameters incorrect.

Authors: We acknowledge that the derivation of the constant 18 is not presented with sufficient explicitness in the current manuscript. In the revised version, we will add a new lemma (likely in Section 3) that provides a step-by-step inequality chain starting from the radius ratios in the Schmidt game, incorporating the specific properties of the δ-filtration on Bad sets, and accounting for the move-size constraints imposed by both the winning and ubiquitously losing strategies. This will confirm the validity of the parameters (1/3, 18δ) and (1/2, 18/δ) or adjust them if necessary. We believe the constant is correct but agree that explicit verification is essential for the claims. revision: yes
Referee: [§6 (dimension estimate)] The upper bound on the Hausdorff dimension of Bad(δ) that is claimed to be strictly less than 1 is derived from the (1/2, 18/δ)-ubiquitously-losing property; because the constant 18/δ itself rests on the unverified estimate above, the dimension statement is conditional on the correctness of that estimate and cannot be regarded as established until the constant is independently confirmed.

Authors: We agree that the dimension upper bound in Section 6 relies on the ubiquitously losing property established in Section 5. Once the explicit chain for the constant 18 is provided as outlined in our response to the previous comment, the dimension estimate will be independently verifiable. We will update Section 6 to reference the new lemma explicitly, ensuring the bound dim_H(Bad(δ)) < 1 is rigorously supported. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit filtration construction and Schmidt-game proofs derive the stated parameters independently.

full rationale

The paper defines the nested filtration Bad(δ) directly from the Diophantine condition on badly approximable numbers, then proves the (1/3,18δ)-winning and (1/2,18/δ)-ubiquitously-losing properties by constructing explicit strategies for the two players in Schmidt games. The constants 18δ and 18/δ are obtained from concrete radius-ratio and approximation-constant bounds inside those strategies, not by fitting to the target statement or by self-citation. The subsequent Hausdorff-dimension bounds and intersection results follow from the general theory of winning/losing sets without reducing to the input filtration by definition. The derivation is therefore self-contained against the external Schmidt-game framework.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper relies on standard background in Diophantine approximation and Schmidt games while introducing new constructions; no free parameters are explicitly fitted to data.

axioms (2)

standard math Standard properties of Hausdorff dimension and Lebesgue measure
Invoked for all dimension bounds and comparisons to full dimension.
domain assumption Definitions and winning/losing criteria of Schmidt games
Central to the claims that Bad(δ) is winning and ubiquitously losing.

invented entities (2)

Filtration Bad(δ) no independent evidence
purpose: Parameterized nested family of badly approximable numbers
New construction introduced to complement the τ-well approximable filtration.
(α, β)-ubiquitously losing sets no independent evidence
purpose: New class of sets in Schmidt games theory
Introduced to obtain the upper dimension bound for Bad(δ).

pith-pipeline@v0.9.0 · 5519 in / 1547 out tokens · 79577 ms · 2026-05-08T05:39:31.730938+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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