Approximate equality for two sums of roots

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.