Density and Defect Distribution of Reduced Ideals Missed by the Reduction Algorithm in Real Quadratic Fields

Source abstract

Schoof asked how many reduced Arakelov divisors cannot arise from the reduction algorithm. Tian and Tran subsequently characterized the image of this algorithm in real quadratic fields as precisely the $1$-reduced ideals. We give a quantitative answer to Schoof's question in the real-quadratic setting by determining the asymptotic proportion of reduced ideals missed by the algorithm. A key arithmetic input is the criterion that a reduced ideal with associated reduced form $(A,b,-r)$ fails to be $1$-reduced if and only if $A-r<b<2(A-r)$. Hence the missed ideals occupy an explicit region of the natural extension of the Gauss map whose relative measure is $λ_{\mathrm{ex}}=\frac{\log(4/3)}{\log4}$. In particular, the missed region occupies exactly half of the digit-one cylinder with respect to the invariant measure. We establish this limiting proportion in two settings. A lattice-point count gives the average asymptotic over fundamental discriminants $Δ\le X$, with error $O(X^{-1/4+\varepsilon})$. For individual real quadratic fields, Duke's equidistribution theorem yields $\operatorname{ECal}(Δ)/κ(Δ) \toλ_{\mathrm{ex}}$ as $Δ\to\infty$. We also determine the limiting distribution of the canonical defect $k/\sqrtΔ$ of the missed ideals. This distribution is supported on $[0,1/\sqrt3-1/2]$, and its lower tail is asymptotic to $\frac{2}{\log(4/3)}t\log(1/t)$ as $t\to0^+$. An exact relation between the integer defect and an endpoint coordinate of the natural extension allows us to transfer equidistribution to the defect law.