On unit fractions with denominators in short intervals (Q2759123)

Erdős and Graham posed a number of questions on the set NEWLINE\[NEWLINEX_k =\{ (x_1, \ldots , x_k): \sum_{j=1}^k \frac{1}{x_j}=1,\quad 0 < x_1 < \ldots < x_k \}.NEWLINE\]NEWLINE In particular they stated problems if one considers elements of \(X_k\) with \(x_1\) as large as possible or \(x_k-x_1\) as small as possible. [See section D11 of \textit{R. K. Guy}, Unsolved problems in number theory. 2nd ed., New York, Springer-Verlag (1994; Zbl 0805.11001).]NEWLINENEWLINENEWLINEHere difficulties arise since a rather dense set \((x_1, \ldots, x_k)\) is still to satisfy the arithmetic divisibility restrictions implied by the definition of \(X_k\). NEWLINENEWLINENEWLINEBy means of smooth numbers, the author overcomes the occurring problems. He proves: For any rational number \(r\) and for all \(N>1\) there exist integers \(x_1, \ldots ,x_k\) with NEWLINE\[NEWLINEN < x_1 < \cdots<x_k\leq\left(e^r + O_r\left(\frac{\log \log N}{\log N} \right)\right) NNEWLINE\]NEWLINE such that NEWLINE\[NEWLINEr = \frac{1}{x_1}+ \ldots + \frac{1}{x_k}.NEWLINE\]NEWLINE Moreover the error term is best possible. The particular case \(r=1\) implies that for infinitely many \(k\) one has \(\max x_1 \sim \frac{k}{e-1}\) and \(\min (x_k-x_1) \sim k\), where the maximum and minimum is taken over all elements \(( x_1, \ldots x_k) \in X_k\).