Generalizations of a result of Jarnik on simultaneous approximation

TL;DR

This paper extends Jarnik's 1930 result on simultaneous approximation of real numbers by integers, providing effective generalizations for powers of real numbers and those in fractal sets like the Cantor set.

Contribution

It offers a new effective generalization of Jarnik's theorem to successive powers and fractal set cases, broadening the scope of simultaneous approximation results.

Findings

Generalized Jarnik's result to powers of real numbers.

Extended approximation results to fractal sets such as the Cantor set.

Provided effective bounds for simultaneous approximation.

Abstract

Consider a non-increasing function $\Psi$ from the positive reals to the positive reals with decay $o(1/x)$ as $x$ tends to infinity. Jarnik proved in 1930 that there exist real numbers $\zeta_{1},...,\zeta_{k}$ together with $1$ linearly independent over $\mathbb{Q}$ with the property that all $q\zeta_{j}$ have distance to the nearest integer smaller than $\Psi(q)$ for infinitely many positive integers $q$, but not much smaller in a very strict sense. We give an effective generalization of this result to the case of successive powers of real $\zeta$. The method also allows to generalize corresponding results for $\zeta$ contained in special fractal sets such as the Cantor set.