Mixed radix numeration bases: Horner's rule, Yang-Baxter equation and Furstenberg's conjecture

TL;DR

The paper addresses how simultaneous decompositions of integers and polynomials in mixed radix bases interact and whether Yang–Baxter structures can illuminate number-theoretic questions such as Furstenberg's conjecture. It develops a two-dimensional rectangle representation, defines local base-changes with $oldsymbol{ ho}_{p,q}$ (and a polynomial analogue $oldsymbol{ ho}_{a,b}$), and proves that triples of bases satisfy the set-theoretic Yang–Baxter equation, connecting to Horner-type algorithms and edge-decorated decompositions. It provides constructive algorithms with complexity $O(l_1l_2)$ for converting between mixed radix bases, reinterprets Horner’s rule and derivatives, and reformulates Furstenberg’s invariant-measure problem in terms of layer-wise Yang–Baxter transformations and Rudolph-like arrays. By bridging number theory with exactly solvable models, the work offers new tools for invariant-measure questions and potential exact results via integrable-model ideas.

Abstract

Mixed radix bases in numeration is a very old notion but it is rarely studied on its own or in relation with concrete problems related to number theory. Starting from the natural question of the conversion of a basis to another for integers as well as polynomials, we use mixed radix bases to introduce two-dimensional arrays with suitable filling rules. These arrays provide algorithms of conversion which uses only a finite number of euclidean division to convert from one basis to another; it is interesting to note that these algorithms are generalizations of the well-known Horner's rule of quick evaluation of polynomials. The two-dimensional arrays with local transformations are reminiscent from statistical mechanics models: we show that changes between three numeration basis are related to the set-theoretical Yang-Baxter equation and this is, up to our knowledge, the first time that such a structure is described in number theory. As an illustration, we reinterpret well-known results around Furstenberg's conjecture in terms of Yang-Baxter transformations between mixed radix bases, hence opening the way to alternative approaches.

Figures (5)

• Figure 1: Two mixed radix bases $\mathbf{b}=(3,3,5,3,5,3,5,5,3)$ and $\mathbf{b}'=(3,3,5,5,3,3,5,5,3)$ for $(p,q)=(3,5)$ that differ by one transposition $\tau_{4,5}$ and the two extremal mixed radix bases $\mathbf{b}^{\mathsf{S}\mathsf{E}}=(3,3,3,3,3,5,5,5,5)$ and $\mathbf{b}^{\mathsf{N}\mathsf{W}}=(5,5,5,5,3,3,3,3,3)$ for lengths $(l_1,l_2)=(5,4)$. Along the edges are written the digits of the decomposition of $n=221$ in all the mixed radix bases of $\mathcal{B}_{3,5}(5,4)$: digits in bold on horizontal lines are base $3$ digits in $\{0,1,2\}$, digits in italic on vertical lines are base 5 digits in $\{0,1,2,3,4\}$.
• Figure 2: Decorations associated to a change of basis from $x=0$ to arbitrary $y\in\mathbb{K}$ of a polynomial of degree $3$. Due to the graded structure of polynomials, only zeroes appear above the dashed line.
• Figure 3: Algorithm \ref{['algo:changebasis']} running on an example with $(p,q)=(3,5)$ and $n=840397253$. Only values computed during the algorithm are present. The input is the lower South line and the output is the left West line. For each vertical strip, the variable $v$ in this algorithm corresponds successively to the values in $\{\mathbf{0},\mathbf{1},\mathbf{2}\}$ associated to the horizontal segements from bottom to top.
• Figure 4: Cube associated to $\mathcal{B}_{p_1,p_2,p_3}(1,1,1)$: the maps $\psi_{p_i,p_j}$ are associated to faces in the dimensions $i$ and $j$. The two extremal paths $\gamma_{p_1,p_2,p_3}$ and $\gamma_{p_3,p_2,p_1}$ are represented respectively by the solid and the dashed lines. On the right, decomposition of $n=59<84=3\cdot 4\cdot 7$ in the mixed radix bases associated to $(p_1,p_2,p_3)=(\mathbf{3},\mathit{4},\mathtt{7})$ (same font for digits on the edges).
• Figure 5: Decomposition of $7/13$ in the mixed radix basis $\mathcal{B}_{3,5,13}=\{3,5,13\}^{\mathbb{N}^*}$. One observes a periodicity of length $3$ in the horizontal dimension associated to $p=3$ (bold digits) and a periodicity of length $4$ in the vertical dimension associated to $q=5$ (italic digits). In the third dimension associated to $13$, all the digits are zero beyond the first ones in the first layer. The periodic horizontal sequence $(\mathsf{7},\mathsf{8},\mathsf{11},\mathsf{7},\ldots)$ corresponds to the orbit of $7/13$ under the map $T_3$. The vertical sequence $(\mathsf{7},\mathsf{9},\mathsf{6},\mathsf{4},\mathsf{7},\ldots)$ corresponds to the orbit of $7/13$ under the map $T_5$. The cube corner on the right is seen from the vertex $C$, in comparison with figure \ref{['fig:cubeforYB']} where it was seen from the point $A$. In particular, there is a transposition of the arguments, indicated by an exponent $t$ between the left and right edge decorations with respect to the arrow of the transformation.

Theorems & Definitions (8)

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• proof : Proof of theorem \ref{['theo:2Drep:numbers']}
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