On the exact Hausdorff dimension of the set of Liouville numbers
The set \(L\) of Liouville numbers -- composed of any irrational \(x\) such that, for all \(n\), there are integers \(p\) and \(q>1\) satisfying \(| x-{p\over q}| <{1\over q^n}\) -- is known to have Hausdorff dimension zero [\textit{J. C. Oxtoby}, ``Measure and category'' (1980; Zbl 0435.28011)]. A translation invariance argument shows that, for any dimension function \(h\), the \(h\)-dimensional Hausdorff measure is either zero or infinity. The main result in the paper amounts to partition the set of concave dimension functions in two classes, namely, if \(h\) approaches zero faster than some power function (specifically \(\limsup_{r\rightarrow 0+}{h(r)\over r^t}=0\) for some \(t>0\)), the \(h\)-dimensional Hausdorff measure of \(L\) is -- of course -- zero; if, on the contrary, \(h\) is a concave dimension function that approaches zero slower than any power function (\(\limsup_{r\rightarrow 0+}{h(r)\over r^t}>0\) for any \(t>0\)), then the \(h\)-dimensional Hausdorff measure of \(L\) is infinity.
scientific article; zbMATH DE number 2156089