We examine a pair of dynamical systems on the plane induced by a pair of spanning trees in the Cayley graph of the Super-Apollonian group of Graham, Lagarias, Mallows, Wilks and Yan. The dynamical systems compute Gaussian rational approximations to complex numbers and are "reflective" versions of the complex continued fractions of A. L. Schmidt. They also describe a reduction algorithm for Lorentz quadruples, in analogy to work of Romik on Pythagorean triples. For these dynamical systems, we produce an invertible extension and an invariant measure, which we conjecture is ergodic. We consider some statistics of the related continued fraction expansions, and we also examine the restriction of these systems to the real line, which gives a reflective version of the usual continued fraction algorithm. Finally, we briefly consider an alternate setup corresponding to a tree of Lorentz quadruples ordered by arithmetic complexity. Quadruples and the Super-Apollonian Group2.1. Simple continued fractions. We begin with a brief overview of simple continued fractions on the real line, described from the perspective and in the language we plan to use for complex continued fractions in the remainder of this paper. The purpose is to provide an explicit analogy for much of what follows.Over the integers, the Euclidean algorithm is the iteration of the division algorithm, which, for a, b ∈ Z, returns q, r ∈ Z so that a = bq + r, 0 ≤ r < |b|. Replacing the pair (a, b) with (b, r), we

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Let (𝑎 𝑛 ) ∞ 𝑛=1 be a lacunary sequence of positive real numbers. Rudnick and Technau showed that for almost all 𝛼 ∈ ℝ, the pair correlation of (𝛼𝑎 𝑛 ) ∞ 𝑛=1 mod 1 is Poissonian. We show that all higher correlations and hence the nearest-neighbour spacing distribution are Poissonian as well, thereby extending a result of Rudnick and Zaharescu to real-valued sequences.