On Unit Fractions with Denominators in Short Intervals
Ernest S. Croot
Abstract
We answer several questions of P. Erdos and R. Graham by showing that for any rational number r > 0, there exist integers n1, n2, ..., nk, k variable, where N < n1 < n2 < ... < nk < (er + or(1) ) N, such that r = 1/n1 + 1/n2 + ... + 1/nk. Moreover, we obtain the best-possible error term for the or(1), which is Or(loglog N/log N).
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