On Unit Fractions with Denominators in Short Intervals

Ernest S. Croot III

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On Unit Fractions with Denominators in Short Intervals

Ernest S. Croot III

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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 < (e^r + o_r(1) ) N, such that r = 1/n1 + 1/n2 + ... + 1/nk. Moreover, we obtain the best-possible error term for the o_r(1), which is O_r(loglog N/log N).

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Ernest S. Croot III. 1999-04-30. On Unit Fractions with Denominators in Short Intervals. https://arxiv.org/abs/math/9904181

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