View source for Gallagher ergodic theorem

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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|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|Lebesgue measure]] either $0$ or $1$.

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

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