Approximate equality for two sums of roots (Q6564670)

For integers \(m\ge 2\) and \(k>s\ge 1\), let \N\[\N r_m(s,k,N):=\min\left|\sum_{i=1}^s {\sqrt[m]{a_i}}-\sum_{i=s+1}^{k} {\sqrt[m]{a_i}}\right|,\quad R_m(s,k,N):=\min\left\|\sum_{i=1}^s {\sqrt[m]{a_i}}-\sum_{i=s+1}^{k} {\sqrt[m]{a_i}}\right\|, \N\]\Nwhere the minimums above are over all the choices of integers \(1\le a_1,\ldots,a_k\le N\). In the case of the quantity \(R_m(s,k,N)\), the author allows \(0\le s\le k\). Using a Liouville-type argument one gets Theorem 1 of the paper which give the inequalities \N\[\Nr_m(s,k,N)\ge \frac{k^{1-m^{k-1}}}{N^{m^{k-2}-1/2}},\quad {\text{and}}\quad R_m(s,k,N)\ge \frac{(2k+1)^{1-m^k}}{N^{m^{k-1}-1/m}}. \N\]\NThe author is interested in the dependence in \(N\) as \(N\) becomes large, so he sets \N\[\Ne_m(s,k):=\limsup_{N\to\infty} \frac{\log(1/r_m(s,k,N))}{\log N}\quad {\text{ and}}\quad E_m(s,k):=\limsup_{N\to\infty} \frac{\log(1/R_m(s,k,N))}{\log N}, \N\]\Nand concludes that \N\[\Ne_m(s,k)\le m^{k-2}-1/m\qquad {\text{and}}\qquad E_m(s,k)\le m^{k-1}-1/m.\N\]\NThe remaining of the paper deals with finding good lower bounds on \(e_m(s,k)\) and \(E_m(s,k)\). Theorem 2 proves that \(E_m(s,k)\ge \min\{2s,k-2,2k-2s\}+2-1/m\). He notes that if \(s=k\) and \(m\) is fixed than the above lower bound does not increase with \(k\). Theorem 3 shows that for \(m\ge 2\), \(E_m(s,k)\ge (k-2)/m+1\), which is better than Theorem 2 when \(s\) is fixed and \(k\) tends to infinity. Theorem 4 gives the values for \N\[\Ne_2(2,4)=E_2(1,3)=E_2(2,3)=7/2.\N\] \NFinally, the author looks at the situation when \(m\) is no longer an integer. That is, for \(\theta>0\) satisfying \(\theta^{-1}\not\in {\mathbb N}\), let \(R_{\theta}(N)\) denote the minimum nonzero value of \(\| a^{1/\theta}\|\) when \(a\in \{1,2,\ldots,N\}\), and let \N\[\NE_{\theta}:=\limsup_{N\to\infty} \frac{\log(1/R_{\theta}(N))}{\log N}. \N\]\NThe author proves that \(E_{1/\theta}=1+\theta(E_{\theta}-1)\) when \(\theta^{\pm 1}\not\in {\mathbb N}\), so it suffices to study \(E_{\theta}\) when \(\theta>1\). Theorem 5 proves some upper and lower bounds for \(E_{\theta}\). Under the \(abc\) conjecture the author proves that \(E_{\theta}\le 1\) whenever \(\theta>1\) is rational. The proofs use linear algebra and calculations with polynomials. The proof of Theorem 5, especially the inequality \(E_{3/2}\ge 1\) uses Danilov's construction of infinitely many pairs of integers \((x,y)\) such that \(0<|x^3-y^2|<0.97 {\sqrt{x}}\).