The twisted Euclidean algorithm: Applications to number theory and geometry

Asymptotic Density of Apollonian-Type Packings

“…In this case, the usual Euclidean algorithm can be applied to R; a simple corollary to this is that if Ep2, Rq is the subgroup of SLp2, Rq generated by upper and lower triangular matrices, then SLp2, Rq " Ep2, Rq. There is a corresponding notion for rings with involutions as well, originally explored by the second author [21]: let pR, σq be a subring with involution of a division algebra; we say that it is σ-Euclidean if there exists a function Φ : R Ñ W to some well-ordered set W such that for all a, b P R with b ‰ 0 and aσpbq P R `, there exists some q P R `such that Φpa ´bqq ă Φpbq. A σ-Euclidean ring has a corresponding σ-Euclidean algorithm, which retains many of the nice properties of the usual Euclidean algorithm; it is easily proved, for example, that if pR, σq is σ-Euclidean, then SL σ p2, Rq " Ep2, Rq, where Ep2, Rq is the subgroup of SL σ p2, Rq generated by upper and lower triangular matrices.…”

Is Unreasonable Slightness a General Phenomenon?

“…A ;-order of pH, ;q is an order O such that O ; " O-equivalently, this is a sub-ring with involution which is also an order. Such an order is called ;-Euclidean [10] if there exists a function Φ : R Ñ W to some well-ordered set W such that for all a, b P O with b ‰ 0 and ab ; P O `, there exists some q P R `such that Φpa ´bqq ă Φpbq. There is a corresponding ;-Euclidean algorithm, which has many of the nice properties as the usual Euclidean algorithm; for example, if pO, ;q is ;-Euclidean, then SL ; p2, Oq " E, where E is the subgroup of SL ; p2, Oq generated by upper and lower triangular matrices [10].…”

“…If it is less than 1, then O is Euclidean (or ;-Euclidean) simply by taking the stathm Φ to be the norm. All such orders have previously been enumerated [10,12], so we are primarily interested in the case where µpΛq " 1. In dim " 3, it is easy to see that there is only one such order, namely Zr ?…”

“…If it is less than 1, then we know that O is ;-Euclidean. If it is exactly 1, then we compute the elements u P H that are distance 1 from O `and note that in each case there exists a maximal ;-order O 1 such that u P O 1 Ą O; these are collected in Table 5 These maximal ;-orders O 1 are ;-Euclidean, and therefore O is semi-;-Euclidean; these ;-orders O are not ;-Euclidean as it is known that O is ;-Euclidean only if O `" O 1`f or some maximal ;-order O 1 [10].…”