Beatty solutions of almost Golomb equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914496043286528 |
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| author | Cloitre, Benoit |
| author_facet | Cloitre, Benoit |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10822 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Beatty solutions of almost Golomb equations Cloitre, Benoit Number Theory 11B83 (primary), 11B37, 39B12, 91A46 (secondary) 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. |
| title | Beatty solutions of almost Golomb equations |
| topic | Number Theory 11B83 (primary), 11B37, 39B12, 91A46 (secondary) |
| url | https://arxiv.org/abs/2604.10822 |