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Correlation between two sequences of irrational numbers

Let us consider the sequence x(n+1) = { + 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

This result follows from section 5.4 in this article as well as results published here. Also, if b1 and b2 are two irrational numbers that are linearly independent over the set of rational numbers, then the correlation between the two sequences (one generated with b1 and the other one with b2) is zero.

But what if b1 = -1 + SQRT(5) / 2 and b2 = 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? The correlation is equal to

But you still need to compute that integral.

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One reader posted the following answer:

It's an application of the Riemann integral criterion for equidistribution for f(x) =  { 5x } * { 4x } applied to the equidistributed sequence n / SQRT(20).


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