# Hyperplane problem in linear congruent generator

From Wikipedia

if an LCG is used to choose points in an n-dimensional space, the points will lie on, at most, $m^{1/n}$ hyperplanes (Marsaglia's Theorem, developed by George Marsaglia). This is due to serial correlation between successive values of the sequence $X_n$.

1. Mathematically, what is the serial correlation between successive values of the sequence $X_n$?

2. I was wondering how the points within one period are distributed to different hyperplanes, as they are generated one after another in a sequence? I guess it is unlikely that the points firstly fill out one hyperplane, and then fill out the next one, and will never visit the hyperplanes visited before in the single period?

Is the time interval between every two consecutive visits to each fixed hyperplane fixed, and same for all hyperplanes? Is it connected to the period of the LCG?

Does taking a small front portion of a sequence of a full period somehow overcome this drawback of LCG? That is the reason why I ask the above questions.

3. Note I think there is a typo. "at most, $m^{1/n}$ hyperplanes" should be "at most, $(n!m)^{1/n}$ hyperplanes". Am I correct?

Thanks!

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