Claims

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1998_11_18_martin: Martin's Theorem 2 gives a k-term representation of 1 with all denominators at most (e/(e-1)+o(1))k, and e/(e-1) < e-1, so for every large k the denominators lie in an interval of width (e-1+o(1))k; the answer is yes.

1999_04_30_croot: Croot's Main Theorem puts a representation of 1 by distinct unit fractions in (N, (e+o(1))N] for every N, so for the infinitely many term counts that occur the width is below (e-1+o(1))k, and about k by the paper's claim.

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