Diophantine approximation with constraints

TL;DR

This paper explores how Dirichlet's theorem on linear forms can be adapted when vectors of coefficients are constrained to form bounded angles with a fixed subspace, revealing that approximation exponents depend solely on the subspace's dimension.

Contribution

It introduces a new framework for Diophantine approximation with angular constraints, providing optimal exponents based only on the subspace dimension.

Findings

Derived best possible approximation exponents depending on subspace dimension

Reduced the problem to a known result of Thurnheer

Constructed a new parametric geometry of numbers with angular constraints

Abstract

Following Schmidt, Thurnheer and Bugeaud-Kristensen, we study how Dirichlet's theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make a bounded acute angle with respect to a fixed proper non-zero subspace $V$ of $\mathbb{R}^n$. Assuming that the point of $\mathbb{R}^n$ that we are approximating has linearly independent coordinates over $\mathbb{Q}$, we obtain best possible exponents of approximation which surprisingly depend only on the dimension of $V$. Our estimates are derived by reduction to a result of Thurnheer, while their optimality follows from a new general construction in parametric geometry of numbers involving angular constraints.