Sums of $k$ unit fractions

Abstract:

Erdős and Straus conjectured that for any positive integer $n\geq 2$ the equation $\frac {4}{n}= \frac {1}{x} + \frac {1}{y} + \frac {1}{z}$ has a solution in positive integers $x, y$, and $z$. Let $m > k \geq 3$ and \[ E_{m,k}(N)= \mid \{ n \leq N \mid \frac {m}{n} = \frac {1}{t_1} + \ldots + \frac {1}{t_k} \text { has no solution with }t_i \in \mathbb {N} \} \mid . \] We show that parametric solutions can be used to find upper bounds on $E_{m,k}(N)$ where the number of parameters increases exponentially with $k$. This enables us to prove \[ E_{m,k}(N) \ll N \exp \left ( -c_{m,k} (\log N)^{1-\frac {1}{2^{k-1}-1}} \right ) \text { with } c_{m,k}>0. \] This improves upon earlier work by Viola (1973) and Shen (1986), and is an “exponential generalization” of the work of Vaughan (1970), who considered the case $k=3$.