# Department of Mathematics

## Diophantine exponents for mildly restricted approximation

Research output: Working paper

We are studying the Diophantine exponent $~\mu_{n,\ell}$ defined for integers $~1 \leq \ell < n$ and a vector $~\alpha \in \mathbb{R}^n$ by letting $~\mu_{n,\ell} = \sup\{\mu \geq 0: 0 < | \underline{x} \cdot \alpha| < H(\ux)^{-\mu} \text{ for infinitely many } \underline{x} \in \mathcal{C}_{n,\ell} \cap \mathbb{Z}^n \}$, where $~ \cdot$ is the scalar product and $~| \cdot |$ denotes the distance to the nearest integer and $~\mathcal{C}_{n,\ell}$ is the generalised cone consisting of all vectors with the height attained among the first $~\ell$ coordinates. We show that the exponent takes all values in the interval $~[\ell+1, \infty)$, with the value $~n$ attained for almost all $~\alpha$. We calculate the Hausdorff dimension of the set of vectors $~\alpha$ with $~\mu_{n,\ell} (\alpha) = \mu$ for $~\mu \geq n$. Finally, letting $~w_n$ denote the exponent obtained by removing the restrictions on $~\underline{x}$, we show that there are vectors $~\alpha$ for which the gaps in the increasing sequence $~\mu_{n,1} (\alpha) \leq \cdots \leq \mu_{n,n-1} (\alpha) \leq w_n (\alpha)$ can be chosen to be arbitrary.