An elementary approach to short interval results for \(k\)-free numbers (Q1106882)

The author proves that for any integer \(k\geq 2\), there is a constant \(c=c(k)\) such that the interval \((x, x+cx^{\theta}]\) contains a \(k\)-free number, where \(\theta =5/(10k+1)\). This improves a result of \textit{H. Halberstam} and \textit{K.-F. Roth} [J. Lond. Math. Soc. 26, 268--273 (1951; Zbl 0043.04901)], who had \(\theta =1/2k+\epsilon.\) In the Halberstam-Roth paper, the problem is reduced to bounding the cardinality of \(S(X,2X)=\{u\in (X,2X]:\) there exists an integer \(m\) such that \(mu^k\in (x,x+h]\}\). The required bounds are obtained from consideration of certain auxiliary polynomials of degree \(k-1\). The author obtains his improvements by bounding the size of \(T(a)=\{u: u\) and \(u+a\) are consecutive elements of \(S(X, 2X)\}\). These bounds are obtained from auxiliary polynomials that are naturally more complicated than those used by Halberstam and Roth. Bounds for \(T(a)\) are then used to obtain improved bounds for \(S(X, 2X)\) for certain ranges of \(X\).