Let us consider the sequence $x(n+1) = \{b+x(n)\}$ with $x(0) = 0$. Here the brackets represent the fractional part function. Thus $x(n)= \{nb\}$ is related to Beatty sequences. If $b$ is irrational, it is known that the numbers $x(n)$ are uniformly distributed, with a lag-$k$ auto-correlation equal to

$6(b+k)^2 + 6(1-2\lfloor b+k+1\rfloor)(b+k)+6\lfloor b+k+1\rfloor^2 -6\lfloor b+k+1\rfloor+1$.

This result follows from section 5.4 in this article as well as results published here. Also, if $b_1$ and $b_2$ are two irrational numbers that are linearly independent over the set of rational numbers, then the correlation between the two sequences (one generated with $b_1$ and the other one with $b_2$) is zero.

But what if $b_1 = -1 + \sqrt{5}/2$ and $b_2 = 2/\sqrt{5}$? In that case, I know with absolute certainty (yet with no proof so far) that the correlation between the two sequences is 1/20 = 0.05. How do you prove this result?


1 Answer 1


Look at the closed subgroup $G\subset \mathbb T^2$ (using the notation that $\mathbb T=\mathbb R / \mathbb Z$) generated by $(b_1,b_2)$; it will be of the form $G=\{ (ax,bx):x\in\mathbb T\} $ for certain integers $a$ and $b$. (In your case, $(a,b)=(5,4)$.) Your orbit $\{(nb_ 1,nb_2):n=1,2,\ldots\}$ is uniformly distributed in $G$. Your correlation is obtained by integrating with respect to the uniform (Haar) measure on $G$, namely by $\int_0^1 (ax\bmod1)(bx\bmod1)dx$.

It might help in general to develop the Fourier series for the functions $ x\mapsto ax\bmod1$ and $bx\mapsto x\bmod1$ on $\mathbb T$.

In your specific case, however, the integral $I= \int_0^1 (5x\bmod1)(4x\bmod1)dx$ is given by the following edge-of-tedious expression $$I=\int_0^{1/5}(5x)(4x)dx+ \int_{1/5}^{1/4}(5x-1)(4x)dx+ \int_{1/4}^{2/5}(5x-1)(4x-1)dx+ \int_{2/5}^{1/2}(5x-2)(4x-1)dx+ \int_{1/2}^{3/5}(5x-2)(4x-2)dx+ \int_{3/5}^{3/4}(5x-3)(4x-2)dx+ \int_{3/4}^{4/5}(5x-3)(4x-3)dx+ \int_{4/5}^{1}(5x-4)(4x-3)dx. $$ The correlation coefficient is $\frac{I-1/4}{1/12}$.

  • $\begingroup$ I meant $-1+(\sqrt{5}/2)$ but in both cases the correlation will be different from zero and the way to compute it is the same. $\endgroup$ May 3, 2019 at 15:52
  • $\begingroup$ I think the solution is $-3+ 12\cdot \lim_{T\rightarrow \infty} \frac{1}{T}\int_0^T \{b_1 x\} \{b_2 x\} dx$, but then you need to compute that integral. I'll check if WolframAlpha can do it. $\endgroup$ May 3, 2019 at 15:55
  • $\begingroup$ Thanks for the result. I haven't checked if it matches my number 1/20, but I would assume it does, and thus I have accepted your answer as the solution. Most importantly, it is your methodology that is interesting, more so than the result. $\endgroup$ May 4, 2019 at 12:59
  • 1
    $\begingroup$ Thank you, but it's really just an application of the en.wikipedia.org/wiki/… for the particular Riemann integrable function $x\mapsto(5x\mod1)(4x\mod1)$ applied to the equidistributed sequence $n/\sqrt{20}$. $\endgroup$ May 4, 2019 at 13:36

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