Looking for Properties of Low Discrepancy Sequences.

[mehr1methanol]

Def: A low discrepancy sequence is a uniformly distributed sequence with minimal discrepancy, O(logN/N).

Question: Let $<x>$ denote the fractal part of an irrational number $x$. Let $(<x_n>)$ be an arbitrary low discrepancy sequence. Is it always true that :
$\lim_{n \to +\infty}|<x_n - x_{n-1}> - <x_{n+1} - x_n>| = 0$

I believe this is an open problem. I'm more than happy to discuss anything that's unclear.

Attempt on the solution:

There are trivial cases such as the followings:

1) $(<n\alpha>)$ where $\alpha$is an irrational number. This is low discrepancy sequence (if you like to see the proof consult me for references). Clearly the conjecture above holds.

2) if I choose the $<x_n>$ such that it's monotonically increasing or monotonically decreasing, then the conjecture above holds because the sequence is also bounded in the unit interval

The difficulty is when if I come up with a sequence $(<x_n>)$, where $<x_n - x_{n-1}>$ alternates between decreasing or increasing. In this case it's not clear to me if the conjecture holds. My goal is to rigorously prove the last case. Or are there cases that I'm missing!??

 

[haruspex]

Def: A low discrepancy sequence is a uniformly distributed sequence with minimal discrepancy, O(logN/N).

Question: Let $<x>$ denote the fractal part of an irrational number $x$. Let $(<x_n>)$ be an arbitrary low discrepancy sequence. Is it always true that :
$\lim_{n \to +\infty}|<x_n - x_{n-1}> - <x_{n+1} - x_n>| = 0$

I believe this is an open problem. I'm more than happy to discuss anything that's unclear.

Attempt on the solution:

There are trivial cases such as the followings:

1) $(<n\alpha>)$ where $\alpha$is an irrational number. This is low discrepancy sequence (if you like to see the proof consult me for references). Clearly the conjecture above holds.

2) if I choose the $<x_n>$ such that it's monotonically increasing or monotonically decreasing, then the conjecture above holds because the sequence is also bounded in the unit interval

The difficulty is when if I come up with a sequence $(<x_n>)$, where $<x_n - x_{n-1}>$ alternates between decreasing or increasing. In this case it's not clear to me if the conjecture holds. My goal is to rigorously prove the last case. Or are there cases that I'm missing!??

Wouldn't the sequence $(<[n/2]\alpha>)$ where [] denotes integer part be low discrepancy?

 

[mehr1methanol]

Wouldn't the sequence $(<[n/2]\alpha>)$ where [] denotes integer part be low discrepancy?

Yes for sure!

I actually solved this problem a while ago! It turns out the conjecture doesn't hold and the counterexample is the van der corput sequence.

 

[mehr1methanol]

I should admit your example is much more clever.