Minkowski’s question mark function is here given as follows: write \(x\in [0,1]\) in its continued fraction representation \(x=[n_1,n_2,\ldots]\) and put \(N_j(x)=\sum_{i=1}^j\,n_i\), then \[ Q(x)=\sum_{j=1}^{\infty}\,(-1)^{j+1}2^{-N_j(x)+1}. \] The author is then interested in the singular-continuous measure \(\mu\), the Minkowski measure, with \(Q(x)\) as its distribution function \[ Q(x)=\int_0^x\,d\mu. \] In §1 (Introduction, 5 pages) the author gives an excellent historical overview of the study of this function and its properties. His research finds compelling evidence (the author’s words) of the precise characterization of \(\mu\) in terms of the conjectures.
Conjecture 1. Minkowski’s measure \(\mu\) is regular in the sense of Ullman-Saff-Totik,
Conjecture 2. Minkowski’s measure \(\mu\) belongs to the Nevai class \(N({1\over 4},{1\over 2})\).
In this Introduction the author introduces 4 connected conjectures and, assuming the validity of Conjecture 1, two propositions concerning the Gaussian weights and Christoffel functions of the orthogonal polynomials following from the weight \(\mu\) and their zeros.
The layout of the paper is as follows:
§1 Introduction: Minkowski’s Q function and its singular measure (\(5\) pages)
§2 Möbius iterated function systems (\(3\) pages)
§3 The Möobius IFS algorithm (\(6{1\over 2}\) pages)
§4 Regularity of Minkowski’s measure (\(1\) page)
§5 Zeros of Minkowski’s polynomials and their regularity (\(2\) pages)
§6 Discrepancy analysis (\(2\) pages)
§7 Gaussian integration of Minkowski’s measure (\(10{1\over 2}\) pages)
§8 The Nevai class of measures (\(4\) pages)
§9 Conclusions (\(1\over 2\) page)
References (\(63\) items)
The sections §3–§8 contain 18 interesting figures of numerical results.