Talk:Diophantine approximation

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Hurwitz Theorem

The article states that: Over the years, this theorem has been improved until the following theorem of Émile Borel (1903). For every irrational number $α$ , there are infinitely many fractions ${\tfrac {p}{q}}\;$ such that

\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}q^{2}}}\,.

I am not an expert, but I think this result is due to Hurwitz, 1891. At least, that's what this reference says: https://www.fmi.uni-sofia.bg/sites/default/files/biblio/fulltext/89-047-058.pdf

Borel, 1903, apparently strengthened this result to say that given three consecutive continued fraction convergents for $α$ , at least one will satisfy the above inequality. At least, that's what the reference I posted says. And I believe that's what the cited theorem in Perron's book says, too, though I don't speak German.

Would someone who knows more about this than I do please take a look, and possibly fix it? Thanks!

Kier07 (talk) 21:45, 26 March 2022 (UTC)[reply]

OK, since no one seems to be rushing to correct the article, I'll do it myself. I will change it to make clear both Hurwitz and Borel's contributions. The Borel part is less essential, and could be removed if that's what people think is best. But we definitely should not be crediting him for Hurwitz's result. Kier07 (talk) 16:56, 29 March 2022 (UTC)[reply]