Beatty solutions of almost Golomb equations

Abstract

The almost Golomb equation of order $r$ is the implicit functional equation $$a\Bigl(\sum_{j=0}^{r-1} a(n{-}j)\Bigr) = n$$ for nondecreasing sequences of positive integers with $a(1)=1$. Its earliest solution, the almost Golomb sequence of order $r$, is $r$-regular in the sense of Allouche and Shallit and has oscillating ratio $a(n)/n$. We prove that for every $r\ge 2$ that is not an even perfect square, the equation admits a second monotone solution given by an inhomogeneous Beatty sequence of slope $1/\!\sqrt{r}$. Composing the equation with $a$ leads to a triple-nested identity which admits a continuous one-parameter family of inhomogeneous Beatty solutions, parametrised by a shift $d$ ranging over an explicit interval. We determine these intervals sharply for $r=2$ and $r=3$, each proved by a local regime analysis combined with equidistribution of an irrational orbit. The endpoints of these intervals sit naturally inside the Pell--Ostrowski framework of Fokkink, and the defect set at the upper endpoint for $r=2$ is characterised as the return-time set of an irrational rotation to an explicit interval.