Waring's problem with Piatetski-Shapiro numbers (Q2810739)

The paper deals with a variant of Waring's problem in which the \(k\)-th powers \(n^k\) are replaced by \(m^k\) (called Piatetski-Shapiro powers) where \(m\) is a number of the form \(\lfloor x^c \rfloor\) for some number \(x\) and a fixed real number \(c>1.\) In order to quote the main result, we put: \(v(4)=96\), \(v(5) = 224\), and for all \(k \geq 3\), \(v(2k) = \frac{2 (3k+1)(9k(k-1)+2)}{k}\) and \(v(2k+1) = 6(3k+2)(3k-1).\) Put also \(B(k) = \frac{k}{2}(\ln(k)+\ln(\ln(k))+2+ O(\frac{\ln(\ln(k))}{\ln(k)})\) and \(C(s,t,k) = \frac{s}{2t(2v(k)-1)+s(v(k)-1)}.\) The main result is:NEWLINENEWLINETheorem. There exists a constant \(K\) such that for all \(k \geq K\) there is a positive integer \(t_0\) bounded above by \(B(k)\) such that, for \(s \geq 1\) and for \(t \geq \lceil \frac{t_0+1}{2} \rceil,\) and for \(c\) such that \(0 < c-1 < C(s,t,k)\) all integers, but a finite number, are sums of at most \(s+2t\) Piatetski-Shapiro powers.NEWLINENEWLINEThe proof uses the Hardy-Littlewood circle method.