Rapid-winning properties for reversed inhomogeneous intersections

Determine whether, for a fixed shift \(\gamma_0\), the intersections \(W^{\gamma_0}(\psi)\cap Bad\) and, more generally, \(W^{\gamma_0}(\psi)\cap\bigcap_{j=1}^r Bad^{\gamma_j}\), admit an analogous rapid-winning property, subject to appropriate arithmetic assumptions on the shifts.

Background

The paper’s proof is asymmetric: the homogeneous lattice produces the ψ\psi-approximation, while the inhomogeneous grid is kept away from the origin to ensure membership in BadγBad^\gamma. The concluding section asks what happens when these roles are reversed, so that an inhomogeneous grid must produce the approximation while the homogeneous lattice and additional forbidden grids remain away from the origin.

The paper notes an arithmetic obstruction: if γj≡kγ0(mod1)\gamma_j\equiv k\gamma_0\pmod 1 and qψ(q)→0q\psi(q)\to0, then Wγ0(ψ)∩BadγjW^{\gamma_0}(\psi)\cap Bad^{\gamma_j} is empty. Thus any positive result must incorporate conditions on the tuple of shifts.

References

It is therefore natural to ask what happens when the roles are reversed. Namely, for a fixed \gamma_0\in\R/\Z$, one may consider

W{\gamma_0}(\psi) := \left{ x\in\R: |qx-\gamma_0|_{\Z}<\psi(q) \ \text{for infinitely many }q \in\N \right}

and ask whether the sets

W{\gamma_0}(\psi)\capBad \qquad\text{or, more generally,}\qquad W{\gamma_0}(\psi)\cap\bigcap_{j=1}rBad{\gamma_j}

admit an analogous rapid-winning property.

— Well and badly approximable sets, and rapid winning  (2608.25349 - Hussain et al., 26 Aug 2026) in Section 6, Concluding remarks