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> Erdős and Graham conjectured that any set that samples some sufficiently large, positive proportion of the whole numbers — it could be 20% or 1% or 0.001% — m
by unbalancedevh 5y ago
> Erdős and Graham conjectured that any set that samples some sufficiently large, positive proportion of the whole numbers — it could be 20% or 1% or 0.001% — must contain a subset whose reciprocals add to 1.
Conjectures like this seems almost like a tautology to me -- If the set doesn't contain such a subset, then by definition it just isn't "sufficiently large," and could be made so simply by expanding it. How is that interesting?
- chalst 5y agoConsider a finite, nonempty set of powers of two that are greater than one. The sum of the reciprocals is between zero and one, and you can always expand any such set to another larger one, which will increase the sum of reciprocals but it will still be under one.
- mauricioc 5y agoThe quoted sentence is an informal description of "positive density", and you can find the definition of density at https://en.wikipedia.org/wiki/Natural_density https://en.wikipedia.org/wiki/Natural_density, or you can take a look at the first page of the source material at https://arxiv.org/pdf/2112.03726.pdf https://arxiv.org/pdf/2112.03726.pdf. In fact, the theorem proved works for a stronger notion called "positive upper density"; the definition of "upper density" is also in the linked Wikipedia page.