No \(b\)-concentrated measures whenever \(b^{nk}(b^ n-1)=1\) (Q1917187)

A locally finite Borel measure \(\mu\) on \(\mathbb{R}\) (\(\mathbb{R}\) -- the real line) is said to be \(b\)-concentrated at the point \(x\in\text{supp}(\mu)\) with \(b>0\) if \[ \limsup_{h\to 0+}\mu((x- bh,x+ bh))/\mu((x- h,x+ h))<b\tag{\(*\)} \] holds. The measure \(\mu\) is said to be \(b\)-concentrated if \((*)\) holds for every \(x\in\text{supp}(\mu)\). Let \(k\) and \(n\) be natural numbers. The author shows that if a positive real number \(b\) satisfies \(b^{nk}(b^n- 1)=1\) then these does not exist any \(b\)-concentrated measure. This improves some of known results of papers of \textit{Z. Buczolich} and \textit{M. Laczkovich} [Acta Math. Hung. 57, No. 3/4, 349-362 (1991; Zbl 0752.28002)] and \textit{Z. Buczolich} [Real Anal. Exch. 19, No. 2, 612-615 (1994; Zbl 0810.28002)]. The full answer on the original question risen by Z. Buczolich and M. Laczkovich gives \textit{G. Drasny} in his paper: ``No \(b\)-concentrated measures whenever \(1< b\leq 2\)'' [Mathematica (in print)].