Additive functions for number systems in function fields
Let \(p\) in \({\mathbb F}_q[X,Y]\) be a separable irreducible polynomial and let \({\mathcal S}={\mathbb F}_q[X,Y]/p{\mathbb F}_q[X,Y]\). A pair \((B,{\mathcal N})\) with \(B\) in \({\mathcal S}\) and \({\mathcal N}\subset {\mathbb F}_q[X]\) is a number system in \({\mathcal S}\) if every \(G\) in \({\mathcal S}\) has a unique finite representation \(G=\sum_{k=0}^{l-1} D_kB^k\) with \(D_k\) in \({\mathcal N}\) and \(D_{l-1}\neq 0\) if \(G\neq 0\). This is the \(B\)-adic representation of \(G\) of length \(L_B(G)=l\). Let \({\mathcal L}_B(m)\) denote the set of all \(G\) in \({\mathcal S}\) with \(B\)-adic length less than \(m\). A function \(f\) defined on \({\mathcal S}\) is \(B\)-additive, or strictly \(B\)-additive, if \(f(G)=\sum_{k=0}^{l-1} f(D_kB^k)\) or \(\sum_{k=0}^{l-1} f(D_k)\) respectively. Now consider \({\mathbb L}={\mathbb F}_q(X,Y)/p{\mathbb F}_q(X,Y)\) and suppose this field extension has degree \(n\) and \({\mathcal S}\) is its ring of integers. For a rational function \(\alpha=A/B\) define the valuation at infinity by \(\nu(\alpha)=\deg B - \deg A\) and let \(\omega\) be the extension of \(\nu\) to the completion \({\mathbb L}_\infty\) of \({\mathbb L}\). Set \(d(\alpha)=-\omega(\alpha)\) to extend the degree to \({\mathbb L}_\infty\). For \({\mathcal T}\subset {\mathbb L}\), set \({\mathcal T}(m)=\{\alpha\;\text{in}\;{\mathcal T}: d(\alpha)\leq m\}\). The authors investigate properties of additive functions which extend the classical results for ordinary number systems to this setting. If the \(f_i\) are \(B_i\)-additive functions on \({\mathcal S}\), the \(B_i\) are coprime and the \(M_i\) are ideals in \({\mathcal S}\), then the function values \(f_i(A)\) with \(A\) in \({\mathcal S}(n)\) are equally distributed amongst their residue classes modulo \(M_i\) and the asymptotic distribution as \(n\rightarrow \infty\) can be described by a central limit theorem. There is a version of Waring's problem in this setting. Every \(N\) in \({\mathcal S}\) with \(d(N)\) sufficiently large can be represented as a sum of \(k\)-th powers, \(N=P_1^k+\cdots+P_s^k\) with \(P_j\) in \({\mathcal S}(\lceil d(N)/k\rceil)\) and \(f_i(P_j)\equiv J_i\bmod M_i\) under a certain assumption relating to a Weyl sum which measures the digital restrictions of the system. The number of representations of \(N\) satisfies a Hardy-Littlewood type asymptotic formula. The proofs rely on properties and estimates of these Weyl sums.
scientific article; zbMATH DE number 5712579
scientific article