Well and badly approximable sets, and rapid winning

@misc{hussain2026well,
  title = {{Well and badly approximable sets, and rapid winning}},
  author = {Mumtaz Hussain and David Simmons},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.25349},
  eprint = {2608.25349},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.25349},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Well and badly approximable sets, and rapid winning
AU  - Mumtaz Hussain
AU  - David Simmons
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2608.25349
UR  - https://arxiv.org/abs/2608.25349
AB  - The set of τ-approximable numbers, 𝒲(τ), has genuinely fractional Hausdorff dimension, whereas the set of inhomogeneously badly approximable numbers, Bad^γ, has full Hausdorff dimension. We determine the Hausdorff dimension of their intersection by introducing the Ψ-rapid game, a scale-sensitive refinement of the rapid game of Hatefi and Simmons (preprint 2024). For every approximation function ψ, we prove that 𝒲(ψ)∩Bad^γ is strong Ψ-rapid winning for a natural gauge Ψ determined by ψ. Unlike Schmidt-type games, whose winning property always implies full Hausdorff dimension, the Ψ-rapid game is calibrated to a prescribed Diophantine scale, so that the resulting dimension bound depends explicitly on the decay of Ψ. In particular, for ψ(q)=q^−τ, τ≥1, we recover the exact Jarník--Besicovitch dimension, that is, HD(𝒲(τ)∩Bad^γ)=2/(τ+1).
ER  -