On functions defined by digits of real numbers (Q1820187)

Let p denote a positive integer, \(p\geq 2\). For \(x\in [0,1)\) let \[ x=\sum^{\infty}_{n=1}\epsilon_ n(x)p^{-n},\quad \epsilon_ n(x)\in \{0,1,...,p-1\} \] be the unique p-base expansion of x (numbers of the form \(ap^{-b}\), \(b\in {\mathbb{N}}\), \(a\in \{0,1,...,p^ b-1\}\), the so-called p-base rational numbers in [0,1), having a finite expansion). The authors introduce and study the set \(M_ p\) of all functions F: [0,1)\(\to {\mathbb{R}}\) for which there exists functions \(f_ n: \{0,1,...,p- 1\}\to {\mathbb{R}}\), \(n\in {\mathbb{N}}\), with the property \(\sum^{\infty}_{n=1}| f_ n(k)| <\infty\) for \(k=0,1,...,p- 1\), such that for every \(x\in [0,1)\), \(F(x)=\sum^{\infty}_{n=1}f_ n(\epsilon_ n(x))\). Further, let \(M^ 0_ p:=\{F\); \(F\in M_ p\), \(F(0)=0\}\). If \(F\in M_ p\) then there exists an \(F^*\in M^ 0_ p\) for which \(F(x)=F^*(x)+F(0).\) The main result of the paper states that if p and q are coprime and \(F\in M_ p\cap M_ q\) then there exist two constants A and B such that \(F(x)=Ax+B\). The case \(p=2\), \(q=3\) is investigated separately and settled elementary. For the general case it is shown that if \(F\in M^ 0_ p\), then F is continuous on \([0,1)\setminus Q_ p\) and right-continuous on \(Q_ p\), where \(Q_ p\) denotes the set of p-base rational numbers in [0,1). Next it is shown that if \(F\in M^ 0_ p\) is continuous on [0,1), then \(F(x)=Ax\). The main result then follows in view of \(Q_ p\cap Q_ q=\{0\}\). The paper ends with a discussion of the necessity of the condition \((p,q)=1\).