具有非零冗余性的数制系统及其在局部复函数理论中的应用
数论
2026-03-31 v1 泛函分析
摘要
本文研究在基 a > 1 a>1 a > 1 A ≡ { 0 , 1 , . . . , r } A\equiv\{0,1,...,r\} A ≡ { 0 , 1 , ... , r } a − 1 < r ∈ N a-1<r\in N a − 1 < r ∈ N x = ∑ n = 1 ∞ α n a n ≡ Δ α 1 α 2 . . . α n . . . r a , α n ∈ A . x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{a^n}\equiv \Delta^{r_a}_{\alpha_1\alpha_2...\alpha_n...}, \alpha_n\in A. x = n = 1 ∑ ∞ a n α n ≡ Δ α 1 α 2 ... α n ... r a , α n ∈ A . [ 0 ; r a − 1 ] [0;\frac{r}{a-1}] [ 0 ; a − 1 r ] Δ c 1 c 2 . . . c m r a = { x : x = Δ c 1 c 2 . . . c m a 1 a 2 . . . a n . . . r a , a n ∈ A } \Delta^{r_a}_{c_1c_2...c_m}= \{x: x=\Delta^{r_a}_{c_1c_2...c_ma_1a_2...a_n...}, a_n\in A\} Δ c 1 c 2 ... c m r a = { x : x = Δ c 1 c 2 ... c m a 1 a 2 ... a n ... r a , a n ∈ A } r a r_a r a f ( x = ∑ n = 1 ∞ α n ( r + 1 ) n ) = Δ α 1 α 2 . . . α n . . . r a , α n ∈ A f\left(x=\sum_{n=1}^{\infty}\frac{\alpha_n}{(r+1)^n}\right)= \Delta^{r_a}_{\alpha_1\alpha_2...\alpha_n...},\alpha_n \in A f ( x = n = 1 ∑ ∞ ( r + 1 ) n α n ) = Δ α 1 α 2 ... α n ... r a , α n ∈ A [ 0 , 1 ] [0,1] [ 0 , 1 ] [ 0 , 1 ] [0,1] [ 0 , 1 ] r = 1 r=1 r = 1 a = 1 + 5 2 a=\frac{1+\sqrt{5}}{2} a = 2 1 + 5 − log a 2 -\log_a2 − log a 2
引用
@article{arxiv.2603.28606,
title = {Numeral systems with non-zero redundancy and their applications in the theory of locally complex functions},
author = {S. O. Vaskevych and Yu. Yu. Vovk and O. M. Pratsiovytyi},
journal= {arXiv preprint arXiv:2603.28606},
year = {2026}
}