A function whose graph is of dimension 1 and has locally an infinite one-dimensional Hausdorff measure (Q2708165)

The author constructs a real-valued, continuous function on \([0,1]\) such that the Hausdorff dimension of its graph is 1, and every subset \(B\) of its graph has infinite 1-dimensional Hausdorff measure if the projection of \(B\) on \([0,1]\) has positive Lebesgue measure. This answers a question posed by \textit{P. Wingren} [Enseign. Math., II. Sér. 41, No. 1-2, 103-110 (1995; Zbl 0834.26006)]. NEWLINENEWLINENEWLINEIn author's example \(f\) is a van der Waerden-like construction: NEWLINE\[NEWLINEf(x)=\sum_{n\geq 1} 2^{-n(n-1)-1} \Delta(2^{n(n+1)+1}x), NEWLINE\]NEWLINE where \(\Delta(y)\) is the distance of \(y\) to the nearest integer. It is also proved that the gauge function of its graph is the function \(\phi(s)=s2^{-2\sqrt{\log_2(1/s)}}\).