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Well and badly approximable sets, and rapid winning

This paper determines the Hausdorff dimension of the intersection between $\tau$ -approximable numbers and inhomogeneously badly approximable numbers by introducing a new scale-sensitive $\Psi$ -rapid game that yields the exact Jarník–Besicovitch dimension of $\frac{2}{\tau+1}$ .

Original authors: Mumtaz Hussain, David Simmons

Published 2026-08-27

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CC BY 4.0

Original authors: Mumtaz Hussain, David Simmons

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of numbers, there is a constant tension between how well a number can be approximated by simple fractions and how stubbornly it resists such simplification. Mathematicians have long studied the "well-approximable" numbers, those that can be closely mimicked by fractions with small denominators, and the "badly approximable" numbers, which maintain a safe distance from all such fractions. For decades, it was known that the set of well-approximable numbers shrinks in size as the approximation becomes more demanding, eventually becoming so sparse that it occupies no length on the number line, yet still possesses a complex, fractional dimension. Conversely, the set of badly approximable numbers is robust, filling the line in a way that suggests it is as large as possible. The natural question that arises is what happens when these two opposing worlds collide: what is the size of the set containing numbers that are both well-approximable to a specific degree and yet stubbornly resistant to approximation in a different, inhomogeneous sense?

This question sits at the heart of a new study by Mumtaz Hussain and David Simmons, who have developed a novel mathematical tool to measure the precise size of this intersection. To understand their achievement, one must first grasp the nature of the sets involved. The "well-approximable" set consists of numbers that get arbitrarily close to fractions infinitely often, while the "badly approximable" set contains numbers that, no matter how hard one tries, cannot be approximated too closely by fractions shifted by a specific amount. While the former set is known to have a dimension that depends on how fast the approximation improves, and the latter is known to be maximally large, their overlap was a mystery. Previous methods could prove that certain sets were large, but they could not distinguish between sets of different fractional sizes; they were too blunt an instrument to measure the subtle, shrinking dimensions of these specific intersections.

Hussain and Simmons solved this by inventing a new type of mathematical game, a refined version of a strategy known as the "rapid game." In this game, two players, Alice and Bob, take turns choosing shrinking intervals on a number line. Bob tries to force the final point of their game into a specific target set, while Alice tries to prevent it. The innovation in this work lies in how the game is played: it is calibrated to a specific scale of approximation. Instead of just asking if a set is large or small, the game is tuned to detect the exact rate at which the intervals shrink. By introducing a "scale-sensitive" rule, the authors created a mechanism where the outcome of the game directly reveals the fractional dimension of the set. If Alice can win this specific, calibrated game, it proves that the set of numbers she is defending is not just large, but has a precise, calculable dimension.

The researchers applied this new game to the intersection of well-approximable numbers and inhomogeneously badly approximable numbers. They demonstrated that for any specific rate of approximation, the set of numbers satisfying both conditions is "winning" in their new game. This victory is not merely a qualitative statement that the set exists; it provides a quantitative formula for its size. The authors proved that the dimension of this intersection is exactly determined by the rate at which the approximation improves. Specifically, if the approximation improves at a certain power law, the dimension of the resulting set is a simple fraction derived from that power. This result confirms a long-held intuition that the faster the approximation requirement, the smaller the set becomes, but it does so with a precision that previous methods could not achieve.

Crucially, the paper rules out the possibility that these sets are empty or trivial in certain cases. The authors show that as long as the shift parameter is not a whole number, the intersection is non-empty and possesses the calculated dimension. They also clarify that if the shift is a whole number, the intersection vanishes for certain rates of approximation, a boundary condition that their framework handles naturally. The confidence in these findings is absolute; the authors provide a rigorous proof that the dimension is exactly the value they calculated, leaving no room for simulation or estimation. They have effectively bridged the gap between the coarse, full-dimensional sets and the fine, fractional-dimensional sets, showing that the tools of game theory can be sharpened to measure the intricate geometry of numbers with unprecedented accuracy.

The implications of this work extend beyond a single formula. By separating the strategy of forcing an approximation from the strategy of maintaining a safe distance, the authors have created a flexible framework. This approach allows them to handle the complex interplay between different types of approximation and avoidance conditions simultaneously. The paper concludes by suggesting that this method could be adapted to more complex scenarios involving multiple shifts or higher dimensions, provided the arithmetic relationships between the shifts are favorable. The work stands as a definitive proof that the intersection of these opposing mathematical worlds is not only real but has a precise, predictable structure, revealing a hidden order in the chaotic distribution of numbers.

Technical Summary: Well and Badly Approximable Sets, and Rapid Winning

Problem Statement
The paper addresses a fundamental question in metric Diophantine approximation: Can a set of genuinely fractional Hausdorff dimension carry a natural "winning" property?

Classically, the set of $\psi$ -approximable numbers, $W(\psi)$ , and the set of badly approximable numbers, $Bad$, represent opposing behaviors.

$W(\psi) = \{x \in \mathbb{R} : \|qx\|_{\mathbb{Z}} < \psi(q) \text{ for infinitely many } q \in \mathbb{N}\}$ . For $\psi(q) = q^{-\tau}$ ( $\tau \ge 1$ ), the Jarník–Besicovitch theorem establishes that $\dim_H W(\tau) = \frac{2}{\tau+1}$ , a value strictly less than 1.
$Bad = \{x \in \mathbb{R} : \inf_{q \ge 1} q\|qx\|_{\mathbb{Z}} > 0\}$ . This set has full Hausdorff dimension ( $\dim_H Bad = 1$ ) and possesses strong largeness properties, specifically being "winning" in the sense of Schmidt games.

The intersection $W(\tau) \cap Bad$ is empty for $\tau > 1$ . However, the inhomogeneous analogue $Bad_\gamma = \{x \in \mathbb{R} : \inf_{q \ge 1} q\|qx - \gamma\|_{\mathbb{Z}} > 0\}$ (for $\gamma \not\equiv 0 \pmod 1$ ) is non-empty and also has full dimension. The central problem is to determine the Hausdorff dimension of the intersection $W(\tau) \cap Bad_\gamma$ . While the upper bound is trivially $\frac{2}{\tau+1}$ (since the set is contained in $W(\tau)$ ), establishing the matching lower bound requires proving that this fractional-dimensional set is "large" in a game-theoretic sense.

Methodology
The authors introduce a new game-theoretic framework called the Strong $\Psi$ -Rapid Game, which serves as a scale-sensitive refinement of the "rapid game" introduced by Hatefi and Simmons.

The Game Structure:
- The game is played on a complete metric space (specifically intervals in $\mathbb{R}$ ).
- Bob chooses nested balls $B_n$ and parameters $\alpha_n$ . Alice chooses sub-balls $A_n \subseteq B_n$ .
- Winning Condition: Alice wins if the intersection point lies in the target set $S$ , or if Bob fails to play " $\Psi$ -rapidly."
- $\Psi$ -Rapidness: Bob is required to play such that $\liminf_{n \to \infty} \frac{\alpha_n}{\Psi(\rho_n)} = 0$ . If this condition fails (i.e., $\liminf_{n \to \infty} \frac{\alpha_n}{\Psi(\rho_n)} > 0$ ), Bob loses by default.
- Unlike classical Schmidt games (where winning implies full dimension), the $\Psi$ -rapid game is calibrated to a specific Diophantine scale. The gauge $\Psi$ is chosen to match the decay of the approximation function $\psi$ .
Dynamical Reformulation:
- The proof utilizes the Dani correspondence, translating the Diophantine problem into the dynamics of flows on the space of unimodular lattices ( $X = SL_2(\mathbb{R})/SL_2(\mathbb{Z})$ ) and grids ( $Y$ ).
- $Bad_\gamma$ corresponds to grid orbits that remain bounded away from the origin (avoiding a cusp).
- $W(\psi)$ corresponds to orbits that enter specific cusp regions (representing good approximations).
Strategic Decomposition:
The authors construct Alice's winning strategy by alternating between two distinct mechanisms:
- The Default Strategy: Ensures the inhomogeneous grid remains uniformly away from the origin (maintaining membership in $Bad_\gamma$ ) and recovers the homogeneous lattice from deep cusp excursions back to a compact region.
- The Auxiliary Strategy: A "one-way" cusp-entry move. Alice forces the homogeneous lattice into a cusp to generate a specific approximation $q$ satisfying $\|qx\| < \psi(q)$ . Crucially, this strategy does not attempt to return the lattice to a bounded region; that task is left to the default strategy in subsequent rounds. This separation eliminates a "square-root loss" present in previous rapid game arguments, allowing for the sharp dimension bound.

Key Contributions and Results

Definition of Strong $\Psi$ -Rapid Winning:
The paper defines a set $S$ as strong $\Psi$ -rapid winning if Alice has a strategy to win the game for any sufficiently small contraction parameter $\beta$ . Theorem 1.9 proves that $W(\psi) \cap Bad_\gamma$ is strong $\Psi_\psi$ -rapid winning, where the gauge $\Psi_\psi$ is derived from $\psi$ via the relation $\Psi_\psi(\rho) \asymp \rho^{-1/2}\psi(\rho^{-1/2})$ .
Dimension Proposition (Proposition 2.1):
The authors prove that if a set $S$ is strong $(\Psi_\omega, \alpha, \beta)$ -rapid winning (where $\Psi_\omega(\rho) = \rho^\omega$ ), then its Hausdorff dimension satisfies:
$\dim_H S \ge \frac{1}{1+\omega}$
This proposition links the decay rate of the gauge function $\Psi$ directly to the lower bound of the Hausdorff dimension.
Main Theorem (Theorem 1.3):
For $\psi(q) = q^{-\tau}$ with $\tau \ge 1$ and $\gamma \not\equiv 0 \pmod 1$ , the paper establishes the exact dimension of the intersection:
$\dim_H (W(\tau) \cap Bad_\gamma) = \frac{2}{\tau + 1}$
This recovers the Jarník–Besicovitch dimension for the intersection, confirming that the "badly approximable" constraint does not reduce the dimension of the well-approximable set in the inhomogeneous case.

Significance and Claims
The paper claims to answer the question of whether genuinely fractional-dimensional Diophantine sets can possess a natural winning property. By introducing the $\Psi$ -rapid game, the authors demonstrate that "winning" can be defined relative to a specific scale, rather than implying full dimension.

Refinement of Existing Theory: The work refines the rapid game of Hatefi and Simmons. The key innovation is the separation of the "approximation" move (forcing entry into a cusp) from the "recovery" move (returning to a bounded region). This separation is explicitly credited with eliminating the square-root loss that previously prevented sharp dimension bounds in rapid game arguments.
Universality: The result holds for any approximation function $\psi$ , with the winning property calibrated to the specific decay of $\psi$ .
Limitations and Future Directions: The authors note that the proof is intrinsically asymmetric (homogeneous lattice for approximation, inhomogeneous grid for avoidance). They identify that reversing these roles (approximating an inhomogeneous shift while avoiding the origin) faces arithmetic obstructions related to the rational dependence of shifts. They suggest that extending this framework to multi-grid settings or higher dimensions requires new "affine cusp-entry lemmas" that can handle uniform lower bounds on determinants involving multiple shifts.

In summary, the paper provides a rigorous game-theoretic characterization of the intersection of well and inhomogeneously badly approximable numbers, proving that this intersection retains the full Jarník–Besicovitch dimension through a novel, scale-sensitive winning strategy.

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