An inhomogeneous Dirichlet theorem via shrinking targets

Abstract

We give an integrability criterion on a real-valued non-increasing function $\psi$ guaranteeing that for almost all (or almost no) pairs $(A,\text{b})$, where $A$ is a real $m\times n$ matrix and $\text{b}\in {\mathbb{R}}^m$, the system $\Vert A\text{q}+\text{b}-\text{p}{\Vert }^m<\psi (T)$, $\Vert \text{q}{\Vert }^n<T$ is solvable in $\text{p}\in {\mathbb{Z}}^m$, $\text{q}\in {\mathbb{Z}}^n$ 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.

22 pages, minor corrections made

References in corpus (2)

• Hausdorff measure of sets of Dirichlet non-improvable numbers
• Shrinking targets problems for flows on homogeneous spaces

Cited by in corpus (4)

• Shrinking targets and eventually always hitting points for interval maps
• Khintchine-type theorems for values of subhomogeneous functions at integer points
• Zero-one laws for eventually always hitting points in in rapidly mixing systems
• Dimension bounds for singular affine forms