No \(b\)-concentrated measures with \(b<1.01\) (Q1332608)

A locally finite Borel measure \(\mu\) on \(R\) (\(R\) -- the real line) is said to be \(b\)-concentrated at the point \(x\in \text{supp}(\mu)\) with a \(b> 0\) if \(\lim_{h\to 0+} \mu((x- bh, x+ bh))/\mu((x- h,x+ h))< b\) holds. Let \(C_ b(\mu)\) be the set of all points where \(\mu\) is \(b\)- concentrated. The measure is said to be \(b\)-concentrated if \(C_ b(\mu)= \text{supp}(\mu)\). The author proves that there is no continuous non-zero \(b\)-concentrated measure with \(b< 1.01\). (This improves some of known results of the paper of the author and \textit{M. Laczkovich} [Acta Math. Hung. 57, No. 3/4, 349- 362 (1991; Zbl 0752.28002)]).