Posted on Sep 20, 2019

# Long-Standing Problem of 'Golden Ratio' and Other Irrational Numbers Solved with 'Magical...

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I obv have a taste for things like this, I realize not all of you do, there's an Arxiv preprint mentioned on the site that's a freebie download, I just found the graph-theoretic proof of the Duffin-Scahefer conjecture here quite novel, honest....

#### Long-Standing Problem of 'Golden Ratio' and Other Irrational Numbers Solved with 'Magical...

Posted from space.com
Posted 25 d ago

Responses: 3

Posted 25 d ago

https://arxiv.org/abs/1907.04593

Here's the site with their graph-theoretic proof, as I'd said, it's a freebie, I obv found this quite clever as an approach, as I'd said....

Here's the site with their graph-theoretic proof, as I'd said, it's a freebie, I obv found this quite clever as an approach, as I'd said....

On the Duffin-Schaeffer conjecture

Let $ψ:\mathbb{N}\to\mathbb{R}_{\ge0}$ be an arbitrary function from thepositive integers to the non-negative reals. Consider the set $\mathcal{A}$ ofreal numbers $α$ for which there are infinitely many reduced fractions$a/q$ such that $|α-a/q|\le ψ(q)/q$. If $\sum_{q=1}^\inftyψ(q)ϕ(q)/q=\infty$, we show that $\mathcal{A}$ has full Lebesgue measure.This answers a question of Duffin and Schaeffer. As a corollary, we alsoestablish a conjecture...

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