Difference between revisions of "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]]) | |
| − | + | $$\left|x-\frac pq\right|<f(q),\quad \gcd(p,q)=1,q>0,$$ | |
| − | The corresponding result, but without the condition | + | 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==== | ====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&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> | + | <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 {{DOI|10.2969/jmsj/01340342}} {{MR|0133297}} {{ZBL|0106.04106}}</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&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> | ||
Latest revision as of 18:14, 11 April 2016
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) |
How to Cite This Entry:
Gallagher ergodic theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gallagher_ergodic_theorem&oldid=17316
Gallagher ergodic theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gallagher_ergodic_theorem&oldid=17316
This article was adapted from an original article by O. Strauch (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article