Erdős Problem #539
Prior state unknown→proved
main exponent determined; sharper subpolynomial factors remain open
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number-theory / Number Theory, Multiplicative Combinatorics
Erdős Problem #539
For $|A| = n$, how small can the cofactor set $Q(A) = \{a / \gcd(a,b) : a, b \in A\}$ be? The answer is $h(n) = n^{1/2 + o(1)}$: a new upper bound $h(n) \le n^{1/2} \exp(O(\sqrt{\log n}))$ matches the classical lower bound.
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For $|A| = n$, how small can the cofactor set $Q(A) = \{a / \gcd(a,b) : a, b \in A\}$ be? The answer is $h(n) = n^{1/2 + o(1)}$: a new upper bound $h(n) \le n^{1/2} \exp(O(\sqrt{\log n}))$ matches the classical lower bound.
main exponent determined; sharper subpolynomial factors remain open
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