Some remarks concerning an example of a minimal, non-uniquely ergodic interval exchange transformation (Q1099279)

This note is a further discussion of the class of examples constructed in a paper by \textit{H. B. Keynes} and \textit{D. Newton} [Math. Z. 148, 101-105 (1976; Zbl 0308.28014)]. A gap in their argumentation is pointed out and then filled. In order to accomplish this correction, the following result is proven. If \(\gamma\) \(\in (0,1)\) has unbounded partial quotients, let \(T_{\gamma}x=x+\gamma (mod 1).\) If \(\beta\) \(\in (0,1)\), let \(f_{\beta}(x)=\chi _{[0,\beta)}(x)-\chi _{[\beta,1)}(x).\) Then \(\gamma\), \(\beta\) can be chosen so that the equation \(g(T_{\gamma}x)=f_{\beta}(x)g(x)\) has a solution. This result relies heavily upon \textit{W. A. Veech} [Trans. Am. Math. Soc. 140, 1-33 (1969; Zbl 0201.056)].