For every alpha>0, is there B(alpha) such that whenever A⊆1,…,N has |A|≥alpha N, some subset S⊆ A satisfies ∑_n∈ S1/n=a/b with b≤ B(alpha)?

Erdős Problem #310 — Bounded denominators from dense reciprocal sets

About 46 years old · traced to 1980

For every $\alpha>0$ , is there $B(\alpha)$ such that whenever $A\subseteq\{1,\ldots,N\}$ has $|A|\geq\alpha N$ , some subset $S\subseteq A$ satisfies $\sum_{n\in S}1/n=a/b$ with $b\leq B(\alpha)$ ?

References

Primary source

P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique 28 (1980).

Additional references

P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique 28 (1980).

number-theory unit-fractions

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