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DIOPHANTINE TRANSFERENCE PRINCIPLE OVER FUNCTION FIELDS
Part of
SOURAV DAS, ARIJIT GANGULY
Journal:

Bulletin of the Australian Mathematical Society / Volume 110 / Issue 2 / October 2024

Published online by Cambridge University Press:

28 February 2024, pp. 216-233

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October 2024
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We study the Diophantine transference principle over function fields. By adapting the approach of Beresnevich and Velani [‘An inhomogeneous transference principle and Diophantine approximation’, Proc. Lond. Math. Soc. (3) 101 (2010), 821–851] to function fields, we extend many results from homogeneous to inhomogeneous Diophantine approximation. This also yields the inhomogeneous Baker–Sprindžuk conjecture over function fields and upper bounds for the general nonextremal scenario.

An inhomogeneous Dirichlet theorem via shrinking targets
Part of
- Ergodic theory
- Diophantine approximation, transcendental number theory
Dmitry Kleinbock, Nick Wadleigh
Journal:

Compositio Mathematica / Volume 155 / Issue 7 / July 2019

Published online by Cambridge University Press:

25 June 2019, pp. 1402-1423

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July 2019
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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
$$\begin{eqnarray}\Vert A\mathbf{q}+\mathbf{b}-\mathbf{p}\Vert ^{m}<\unicode[STIX]{x1D713}(T),\quad \Vert \mathbf{q}\Vert ^{n}<T,\end{eqnarray}$$
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.

A NOTE ON INHOMOGENEOUS DIOPHANTINE APPROXIMATION IN BETA-DYNAMICAL SYSTEM
Part of
- Classical measure theory
- Probabilistic theory: distribution modulo $1$; metric theory of algorithms
YUEHUA GE, FAN LÜ
Journal:

Bulletin of the Australian Mathematical Society / Volume 91 / Issue 1 / February 2015

Published online by Cambridge University Press:

12 September 2014, pp. 34-40

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February 2015
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We study the distribution of the orbits of real numbers under the beta-transformation $T_{{\it\beta}}$ for any ${\it\beta}>1$. More precisely, for any real number ${\it\beta}>1$ and a positive function ${\it\varphi}:\mathbb{N}\rightarrow \mathbb{R}^{+}$, we determine the Lebesgue measure and the Hausdorff dimension of the following set:
$$\begin{eqnarray}E(T_{{\it\beta}},{\it\varphi})=\{(x,y)\in [0,1]\times [0,1]:|T_{{\it\beta}}^{n}x-y|<{\it\varphi}(n)\text{ for infinitely many }n\in \mathbb{N}\}.\end{eqnarray}$$