A mathematical preprint reports an exact Hausdorff-dimension formula for a set of real numbers that must be both well approximable and inhomogeneously badly approximable. In the stated nonzero-shift setting, the paper gives dimH(W(τ) ∩ Badγ) = 2/(τ+1) when γ is not 0 and τ is at least 1. Hausdorff dimension is a way to measure sets that may be much thinner than an ordinary interval.
The work is theoretical. It studies sets of real numbers defined by approximation functions and inhomogeneous avoidance conditions, rather than an empirical sample. No participants, observations or sampled records were used.
A game tuned to the scale of approximation
The central tool is a strong Ψ-rapid game, a framework whose moves are calibrated to the scale demanded by an approximation function ψ. The main theorem states that W(ψ) ∩ Badγ is strong Ψψ-rapid winning on every nonempty interval, for γ in R/Z and every approximation function ψ. In practical terms, the theorem identifies a strategy that keeps the target set in play across the relevant scales.
The calibration is specific. The gauge Ψψ is built from the infimum of qψ(q) over a fixed proportional window of denominators and is evaluated at scale ρ−1/2. For the power-law choice ψ(q)=q−τ, with τ at least 1, the natural gauge satisfies Ψψ(ρ) ≍ ρ(τ−1)/2.
The game also supplies a dimension lower bound. When Ψ is comparable to ρ^ω and the set is strong rapid winning for every sufficiently small β, the construction gives dimH S ≥ 1/(1+ω). This is the dimension estimate that supports the paper’s broader use of rapid games for fractional-dimensional sets.
How one strategy meets both demands
The proof translates game balls into trajectories of unimodular lattices and inhomogeneous grids under dynamical flows. A dynamical correspondence links homogeneous bad approximability to bounded lattice orbits, while the inhomogeneous condition becomes eventual avoidance of the origin by the relevant grids.
The strategy alternates between default and auxiliary moves. Default recovery restores boundedness and maintains avoidance of the origin by the inhomogeneous grid. Auxiliary moves make a one-way entry into a cusp, the part of the construction that creates unusually strong approximation. At a ball of radius ρ, such a move produces a denominator q on the scale ρ−1/2 and an approximation error bounded by the stated q−1α scale.
With infinitely many auxiliary moves, the denominators tend to infinity and the outcome meets the required well-approximation condition. The grid-avoidance part of the strategy maintains the inhomogeneous badness condition at the same time. The resulting point therefore belongs to both W(ψ) and Badγ.
What the result does not settle
The document uses different shift conditions in two central statements. The winning theorem is stated for γ in R/Z, while the exact dimension formula is stated for γ in (R/Z) \ {0} and τ at least 1. The supplied analysis does not reconcile that difference, so the formula should be read within its stated nonzero-shift setting.
The reversed inhomogeneous problem is not solved in general. If, for an integer k at least 1, the shifts satisfy γj ≡ kγ0 modulo 1 and qψ(q) tends to 0, the reversed intersection is empty. The arithmetic relation can therefore obstruct a direct reversal of the argument.
The preprint points to multi-grid extensions and to higher-dimensional, weighted and systems-of-linear-forms settings as future work. Those directions are not established by the present result.
The supplied document is an arXiv version 1 preprint dated 26 August 2026. It contains no funding statement.
Paper data and sources
Original title: Well and badly approximable sets, and rapid winning
Authors: Mumtaz Hussain, David Simmons
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text