An Erd\H{o}s-Wintner theorem for second-order linear recurrent bases

Abstract

Let $a,b$ be integers with $1\le b\le a$, and let \[ G_0=1,\qquad G_1=a+1,\qquad G_{n+2}=aG_{n+1}+bG_n. \] For real-valued functions which are additive in the greedy $G$-digits, we prove a necessary-and-sufficient criterion for the existence of a limiting distribution. The criterion consists of a first-order drift series and a quadratic digit-energy series, and it recovers the Zeckendorf theorem when $a=b=1$. The main difficulty is necessity: a one-step transfer matrix detects all non-maximal digit values, while the maximal digit becomes visible only in a two-step product. A weighted Euclidean norm symmetrizes the untwisted companion matrix and makes both contractions occur at the true Perron scale. Sufficiency follows from a two-dimensional Perron product lemma with square-summable transverse perturbations and a convergent, not necessarily absolutely convergent, Perron drift. The limiting characteristic function admits a scalar infinite-product representation in a neighbourhood of the origin and a global matrix-product representation at arbitrary frequencies.