Gallagher ergodic theorem

Let $f(q)$ be a non-negative function defined on the positive integers. Gallagher's ergodic theorem, or Gallagher's zero-one law states that the set of real numbers $x$ in $0\leq x\leq1$ for which the Diophantine inequality (cf. also Diophantine equations)

$$\left|x-\frac pq\right|<f(q),\quad \gcd(p,q)=1,q>0,$$

has infinitely many integer solutions $p$, $q$ has Lebesgue measure either $0$ or $1$.

The corresponding result, but without the condition $\gcd(p,q)=1$, was given by J.W.S. Cassels [a1]. P. Gallagher [a2] established his result for dimension one using the method of Cassels. The $k$-dimensional generalization is due to V.T. Vil'chinskii [a5]. A complex version is given in [a3].

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