Approximate equality for two sums of roots

Abstract

In this paper, we consider the problem of finding how close two sums of mth roots can be to each other. For integers $m\ge 2$, $k\ge 1$ and $0\le s\le k$, let $e_m(s,k)>0$ and $E_m(s,k)>0$ be the largest exponents such that for infinitely many integers N there exist k positive integers $a_1,\dots ,a_k\le N$ for which two sums of their mth roots ${\sum }_{j=1}^s\sqrt[m]{a_j}$ and ${\sum }_{j=s+1}^k\sqrt[m]{a_j}$ are distinct but not further than $N^{-e_m(s,k)}$ from each other, or they are distinct modulo 1 but not further than $N^{-E_m(s,k)}$ from each other modulo 1. Some upper bounds on $e_m(s,k)$ and $E_m(s,k)$ can be derived by a Liouville-type argument, while lower bounds are usually difficult to obtain. We prove that $e_m(s,k)\ge \min (2s,k-1,2k-2s)-1/m$ for $1\le s<k$ and that $E_m(s,k)\ge \min (2s,k-2,2k-2s)+2-1/m$ for $0\le s\le k$. Very recently, Steinerberger managed to show that $E_2(k,k)\ge c\sqrt[3]{k}$, where $c>0$ is a small absolute constant. This seems to be the first result when the bound for $s=k$ is increasing in k. By an entirely different argument, for any integers $m\ge 2$, $k\ge 1$ and s in the range $0\le s\le k$, we show that $E_m(s,k)\ge (k-2)/m+1$. In particular, for $m=2$ and any non-negative integer $s\le k$, this yields the bound $E_2(s,k)\ge k/2$, which is much better than $c\sqrt[3]{k}$. We also prove that $e_2(2,4)=7/2$, which settles a problem raised by O'Rourke in 1981. These problems can be also considered for non-integer m. In particular, we show that $1\le E_{3/2}(1,1)\le 4/3$, and that $E_{3/2}(1,1)=1$ under assumption of the abc-conjecture.