Beatty solutions of almost Golomb equations

TL;DR

This paper investigates solutions to the almost Golomb equation, revealing that for most orders, there exist multiple monotone solutions including Beatty sequences, and provides a detailed analysis of their properties and parameter intervals.

Contribution

It introduces new monotone solutions to the almost Golomb equation using inhomogeneous Beatty sequences and characterizes their parameter intervals with sharp analysis for specific cases.

Findings

Existence of a second monotone solution for most orders r

Explicit characterization of solution intervals for r=2 and r=3

Connection of solution endpoints with Pell–Ostrowski framework

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.