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An inhomogeneous Dirichlet theorem via shrinking targets

  • Dmitry Kleinbock (a1) and Nick Wadleigh (a2)

Abstract

We give an integrability criterion on a real-valued non-increasing function $\unicode[STIX]{x1D713}$ guaranteeing that for almost all (or almost no) pairs $(A,\mathbf{b})$ , where $A$ is a real $m\times n$ matrix and $\mathbf{b}\in \mathbb{R}^{m}$ , the system

$$\begin{eqnarray}\Vert A\mathbf{q}+\mathbf{b}-\mathbf{p}\Vert ^{m}<\unicode[STIX]{x1D713}(T),\quad \Vert \mathbf{q}\Vert ^{n}<T,\end{eqnarray}$$
is solvable in $\mathbf{p}\in \mathbb{Z}^{m}$ , $\mathbf{q}\in \mathbb{Z}^{n}$ for all sufficiently large  $T$ . The proof consists of a reduction to a shrinking target problem on the space of grids in $\mathbb{R}^{m+n}$ . We also comment on the homogeneous counterpart to this problem, whose $m=n=1$ case was recently solved, but whose general case remains open.

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The first-named author was supported by NSF grants DMS-1101320 and DMS-1600814.

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References

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An inhomogeneous Dirichlet theorem via shrinking targets

  • Dmitry Kleinbock (a1) and Nick Wadleigh (a2)

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