On a generalization of Farey sequences. II (Q1903500)

The authors consider a generalisation of Farey sequences of rationals to quadratic irrationals. Given \(N\in\mathbb{N}\), the set of coprime triples \(h=(a,b,c)\) of integers satisfying (i) \(| a|\), \(| b|\), \(| c|\leq N\); (ii) either \(c>0\), \(b^2-4 ac>0\) and square-free, or \(c=0\) and \(b>0\), (iii) the larger root \(\alpha\) of \(a-bx+ cx^2=0\) lies in \([0,1]\); and ordered by the size of \(\alpha\), is called the Haros sequence of order \(N\). The case \(c=0\) corresponds to the Farey sequence. Haros proved that successive fractions \(p/q, p'/q'\) in a Farey sequence satisfy \(pq'- p'q=1\) (the quotation from \textit{G. H. Hardy}'s A Mathematician's Apology [first edition (1940; Zbl 0025.19301)] about the inappropriateness of Farey's name being associated with the sequence is on p. 21). \textit{H. Brown} and \textit{K. Mahler} [J. Number Theory 3, 364-370 (1971; Zbl 0221.10014)] found that for small values of \(N\), the determinant \(\Delta_t\) of successive triples \(h_{t-1}, h_t, h_{t+1}\) in the corresponding Haros sequence is at most 1 whenever the roots of the three corresponding quadratics \(a-bx+cx^2\) are irrational. The authors prove that Haros's result holds `on average' for the determinant \(\Delta_t\) in the sense that \[ \sum_{t\in T} |\Delta_t |=O (\text{card} {\mathcal T}), \] where \({\mathcal T}\) is the set of indices for which the coefficients \(c_{t-1}, c_t, c_{t+1} >0\). The proof involves geometric and number theoretic arguments. Part I appeared in [A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math. 143, 243-246 (1993; Zbl 0791.11009)].