AI 中文总结
针对二阶线性递推基的贪心G-进制数位可加实值函数，本文证明了其极限分布存在的充要准则，解决了最大数位仅在两步乘积中显现的难点，推导了极限特征函数的乘积表示。
AI 中文摘要
设a、b为满足1≤b≤a的整数，定义G₀=1，G₁=a+1，Gₙ₊₂=aGₙ₊₁+bGₙ。对于在贪心G-进制数位上可加的实值函数，本文证明了其存在极限分布的充要准则，该准则由一阶漂移级数和二次数位能量级数构成，当a=b=1时可还原为Zeckendorf定理。主要难点在于必要性：单步转移矩阵可检测所有非最大数位值，而最大数位仅在两步乘积中显现；加权欧几里得范数将非扭转伴随矩阵对称化，使两者收缩均发生在真实Perron尺度上。充分性源于二维Perron乘积引理，该引理具备平方可积横向扰动和收敛（未必绝对收敛）的Perron漂移。极限特征函数在原点邻域具有标量无穷乘积表示，在任意频率处具有全局矩阵乘积表示。
英文摘要
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.
Comments19 pages
