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An inhomogeneous Dirichlet theorem via shrinking targets
- Part of
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- Journal:
- Compositio Mathematica / Volume 155 / Issue 7 / July 2019
- Published online by Cambridge University Press:
- 25 June 2019, pp. 1402-1423
- Print publication:
- July 2019
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- Article
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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 is solvable in
$$\begin{eqnarray}\Vert A\mathbf{q}+\mathbf{b}-\mathbf{p}\Vert ^{m}<\unicode[STIX]{x1D713}(T),\quad \Vert \mathbf{q}\Vert ^{n}<T,\end{eqnarray}$$
$\mathbf{p}\in \mathbb{Z}^{m}$, 
$\mathbf{q}\in \mathbb{Z}^{n}$ 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.
