Of course the $X_k$ are not independent as random variables. So I assume you are referring to some notion of asymptotic independence, and it would help if you state your conjecture more precisely. One natural guess is equidistribution (see [1]) of r-tuples $(X_k,\ldots,X_{k+r-1})$. However, The triples $(X_k, X_{k+1},X_{k+2})$ will lie on a bounded number of planes in $[0,1]^3$; there will be at most 8 such planes in your example. This can be seen if one graphs these triples in 3D. A similar problem arose in the classical RANDU random number generator, see [3].

[1] Kuipers, L.; Niederreiter, H. (2006) [1974]. Uniform Distribution of Sequences. Dover Publications. [2] https://en.wikipedia.org/wiki/Equidistributed_sequence# [3] https://en.wikipedia.org/wiki/RANDU

Of course the $X_k$ are not independent as random variables. So I assume you are referring to some notion of asymptotic independence, and it would help if you state your conjecture more precisely. One natural guess is equidistribution (see [1]) of r-tuples $(X_k,\ldots,X_{k+r-1})$. However, The triples $(X_k, X_{k+1},X_{k+2})$ will lie on a bounded number of planes in $[0,1]^3$; there will be at most 9 such planes in your example. This can be seen if one graphs these triples in 3D. A similar problem arose in the classical RANDU random number generator, see [3].

[1] Kuipers, L.; Niederreiter, H. (2006) [1974]. Uniform Distribution of Sequences. Dover Publications. [2] https://en.wikipedia.org/wiki/Equidistributed_sequence# [3] https://en.wikipedia.org/wiki/RANDU