On some questions of Erdős and Graham about Egyptian fractions

Abstract

In this paper it is proved that, for x sufficiently large, every integer m with 1⩽m⩽[∑1⩽n⩽x1n−92(log log x)2(1+o(1))log x] can be written as m = Σ1≤n≤xεn/n, where εi, = 0 or 1.

On the Number of Integers Representable as Sums of Unit Fractions, III

“…Then recently, in [2], Croot improved our previous results so that log n þ g À 9 2 þ oð1Þ ðlog 2 nÞ 2 log n 4jNðnÞj4log n þ g À 1 2 þ oð1Þ ðlog 2 nÞ 2 log n :…”

On a question of Erdös and Graham

“…Using (3) we derive (#W (X )) k 2 p − 2 k+1 m 2k X 2k−k/2m 2 м (#W (X )) k 2 p − 2 k+1 4 k/m 2 m 2k X 2k p −k/2m 2 (2m− 1) м (#W (X )) k 2 p − 2 k+1 4 k/m 2 m 2k X 2k p −1−1/2m 2 (2m−1) . Now from the prime number theorem we see that for sufficiently large X the last expression is positive and the desired result follows.…”

“…Croot [1] has recently proved that one can select k Ϲ log 3+o (1) p pairwise distinct integers in the interval [1, p ε ] that satisfy (1). Here we give a positive answer to the above question of Erdős and Graham.…”

On unit fractions with denominators in short intervals

“…then none of the x i 's can be divisible by a prime p > x k / log x k (this idea appears in [2], [3], and [6]). It will turn out that this forces…”

“…The proof of this Proposition rests on a highly technical corollary to a lemma taken from an earlier paper by the author (see [2]). For completeness, we will prove both this lemma and its corollary in section II of the paper.…”