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.

 

[blue_raver22]

I would approach this problem by first looking at the existing literature on low discrepancy sequences and their properties. There have been numerous studies and research on this topic, and it is important to review and understand the current state of knowledge before attempting to solve the problem.

After reviewing the literature, I would start by examining the definition of low discrepancy sequences and their properties. It is important to fully understand what constitutes a low discrepancy sequence and how it differs from other types of sequences.

Next, I would analyze the conjecture given in the question and try to understand its implications and potential limitations. It is possible that there are certain assumptions or conditions that need to be met for the conjecture to hold, and it is important to identify these in order to find a solution.

I would also consider different approaches for proving the conjecture. This could involve using mathematical proofs, simulations, or even conducting experiments to gather data and analyze it.

Additionally, I would seek the help of other experts in the field, such as mathematicians or other scientists who have experience with low discrepancy sequences. Collaborating with others can often lead to new insights and ideas for solving a problem.

In conclusion, solving this problem would require a thorough understanding of low discrepancy sequences and their properties, as well as careful analysis and collaboration with others. It may also require innovative thinking and new approaches to proving the conjecture.