Sums of \(k\) unit fractions (Q2716136)

The Erdős-Strauss [-Schinzel] conjecture states the solvability of the Diophantine equation NEWLINE\[NEWLINE {4\over n} = {1 \over x} + {1 \over y} + {1 \over z} .NEWLINE\]NEWLINE The author's paper deals with upper estimates of the number \(E_{m,k} (N) \) of [exceptional] integers \( n \leq N \) for which there is no Diophantine representation NEWLINE\[NEWLINE {m\over n} = {1\over t_1} + \dots + {1\over t_k}, t_\kappa \in { \mathbb N}.NEWLINE\]NEWLINE The author proves the estimate NEWLINE\[NEWLINE E_{m,k}(N) \ll {N \over \exp \left\{ C_{m,k} \cdot \left(\log N^{1-{1 \over 2^{k-1}-1}}\right) \right\} } NEWLINE\]NEWLINE for \( k \geq 3 \) (and \( m > k\)). For \( k = 3 \) this result does not improve on \textit{R. C. Vaughan}'s result [Mathematika 17, 193-198 (1970; Zbl 0219.10023)], but for \(k\geq 4\) it does improve considerably upon estimates of \textit{C. Viola} [Acta Arith. 22, 339-352 (1973; Zbl 0266.10016)] and \textit{Z. Shen} [in Chin. Ann. Math., Ser. B 7, 213-220 (1986; Zbl 0596.10017)].NEWLINENEWLINENEWLINEOf course, the author makes use of the ``Large Sieve'', as Vaughan did already. But his better estimates (for \( k \geq 4\)) are due to clever improvements in the combinatorial part of the proof (and to a skillful notation). The author is able to make use of \( 2^{k-1}\) ``parameters'' instead of \( k \) resp. \( (k+1)\) parameters used by Viola resp. Shen. This helps in finding ``many'' residue classes such that NEWLINE\[NEWLINE {m\over n} = {1\over t_1} + \dots + {1\over t_k} NEWLINE\]NEWLINE is solvable, and these many residue classes make the ``Large Sieve'' more effective.