# Low discrepancy sequences failing Poissonian pair correlations

## TL;DR

This paper demonstrates that certain well-distributed sequences, despite having minimal discrepancy, do not exhibit Poissonian pair correlations, challenging assumptions about their randomness properties.

## Contribution

It proves that sequences with minimal discrepancy, including Levin's number and variants, fail to have Poissonian pair correlations, revealing limitations in their randomness.

## Key findings

- Sequences with minimal discrepancy do not have Poissonian pair correlations.
- Levin's number and its variants fail to exhibit Poissonian pair correlations.
- Challenges assumptions linking low discrepancy with Poissonian pair correlations.

## Abstract

M. Levin defined a real number $x$ that satisfies that the sequence of the fractional parts of $(2^n x)_{n\geq 1}$ are such that the first $N$ terms have discrepancy $O((\log N)^2/ N)$, which is the smallest discrepancy known for this kind of parametric sequences. In this work we show that the fractional parts of the sequence $(2^n x)_{n\geq 1}$ fail to have Poissonian pair correlations. Moreover, we show that all the real numbers $x$ that are variants of Levin's number using Pascal triangle matrices are such that the fractional parts of the sequence $(2^n x)_{n\geq 1}$ fail to have Poissonian pair correlations.