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].

References

[a1]	J.W.S. Cassels, "Some metrical theorems of Diophantine approximation I" Proc. Cambridge Philos. Soc. , 46 (1950) pp. 209–218
[a2]	P.X. Gallagher, "Approximation by reduced fractions" J. Math. Soc. Japan , 13 (1961) pp. 342–345 DOI 10.2969/jmsj/01340342 MR0133297 Zbl 0106.04106
[a3]	H. Nakada, G. Wagner, "Duffin–Schaeffer theorem of diophantine approximation for complex number" Astérisque , 198–200 (1991) pp. 259–263
[a4]	V.G. Sprindzuk, "Metric theory of diophantine approximations" , Winston&Wiley (1979) (In Russian)
[a5]	V.T. Vil'chinskii, "On simultaneous approximations by irreducible fractions" Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk (1981) pp. 41–47 (In Russian)