Erdos-Graham Question on Averages of Unit Fractions
Statement
Erdos and Graham asked whether a positive-density subset of can avoid having any two distinct elements whose unit fractions average to a unit fraction. It can: there is a constant such that for all large some of size has that property, which also gives the best known lower bounds for related unit-fraction avoidance problems.
Record
Comments
No person has examined this. Nothing here has been checked at all. say whether it holds →
construction · #1
Will Sawin, using ChatGPTThat credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.
The author gives a narrative rather than a blanket acknowledgement. Starting from a computation of Stijn Cambie, he asked ChatGPT to look for patterns in Cambie's extremal set that might suggest a generalization; it observed that in a pair with a given ratio the larger element is usually absent unless the smaller is absent for other reasons, and described a change of variables. Dropping the hedges in that observation gives the set the paper analyzes, which turns out to be essentially where Hooley's function takes its minimum value. The model was also used for reference search and proofreading.
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