The convergence of the generalised Selmer algorithm (Q891184)

For positive integers \(a,b\) and \(d=a+b\), let \(X_d\) be the space of sorted \(d\)-tuples \(\vec x =(x_1,\dots,x_d)\) with \(0\leq x_1\leq x_2 \leq \dots \leq x_d\). The map \(F_{a,b}:X_d\to X_d\) is defined by \[ F_{a,b}(x_1,\dots,x_a,x_{a+1},\dots x_{a+b})=\mathrm{sort}(x_1,\dots,x_a,x_{a+1}-x_1,\dots,x_{a+b}-x_1), \] where the sorting rearranges the coordinates into increasing order. (\textit{F. Schweiger} [Multidimensional continued fractions. Oxford: Oxford University Press (2000; Zbl 0981.11029)] has coined the term ``subtractive algorithm'' for maps like \(F_{a,b}\).) If we iterate \(F_{a,b}\) at an arbitrary initial point \(\vec x \in X_d\) then the limit \(\vec x^\infty=\lim_{k\to \infty}F_{a,b}^k(\vec x)\) exists by monotonicity and it is a fixed point of \(F_{a,b}\) by continuity. Therefore, the first coordinate of \(\vec x^\infty\) is equal to zero. How many more zero coordinates can we expect? The first main result of the paper concerns the number of zeroes in the limit \(\vec x^\infty\): Theorem 1. Let \(\vec x^\infty=\lim_{k\to \infty}F_{a,b}^k(\vec x)\) for an ordered \(d\)-tuple \(\vec x\in X_d\). The ascending chain \(\mathcal P_1 \subset \mathcal P_2 \subset \dots \subset \mathcal P_d\) is defined by \[ \mathcal P_r=\{\vec x \in X_d: \vec x_r^\infty >0\}. \] Then \(\mathcal P_1= \emptyset\) and \(\mathcal P_r\) is a null set if \(r \leq a+1\). The first element of the chain that has positive measure is \(\mathcal P_{a+2}\). If \(r \leq \min\{d,2a\}\), then the complement of \(\mathcal P_r\) in \(X_d\) has positive measure. According to the authors' conjecture (supported by numerical experiments), the complement of \(\mathcal P_r\) in \(X_d\) is a null set provided \(r>2a\). Note that if \(a=b=1\) then the subtractive algorithm \(F_{a,b}\) is a homogeneous version of the Farey map. More generally, if \(b=1\), then the map \(F_{a,b}\) is known as Selmer's algorithm since it was first considered by Selmer.