Low discrepancy sequences failing Poissonian pair correlations

2023

Aequat. Math.

For a finite sequence $$(x_i)_{i=1}^N$$ ( x i ) i = 1 N in the unit interval, we introduce the gap ratio function which measures the size of the maximal gap length relative to all other gap lengths. This function (asymptotically) captures a lot of information about the degree of uniformity of the sequence. We discuss connections to the theories of dispersion, discrepancy, pair correlation statistics and covering numbers. Furthermore, we explicitly calculate the gap ratio function for some important classes of uniformly distributed sequences.

Deviation from equidistance for one-dimensional sequences

2023

Aequat. Math.

Some connections between discrepancy, finite gap properties, and pair correlations

2022

Monatsh Math

A generic uniformly distributed sequence $$(x_n)_{n \in \mathbb {N}}$$ ( x n ) n ∈ N in [0, 1) possesses Poissonian pair correlations (PPC). Vice versa, it has been proven that a sequence with PPC is uniformly distributed. Grepstad and Larcher gave an explicit upper bound for the discrepancy of a sequence given that it has PPC. As a first result, we generalize here their result to the case of $$\alpha $$ α -pair correlations with $$0< \alpha < 1$$ 0 < α < 1 . Since the highest possible level of uniformity is achieved by low-discrepancy sequences it is tempting to assume that there are examples of such sequences which also have PPC. Although there are no such known examples, we prove that every low-discrepancy sequence has at least $$\alpha $$ α -pair correlations for $$0< \alpha <1$$ 0 < α < 1 . According to Larcher and Stockinger, the reason why many known classes of low-discrepancy sequences fail to have PPC is their finite gap property. In this article, we furthermore show that the discrepancy of a sequence with the finite gap property plus a condition on the distribution of the different gap lengths can be estimated. As a concrete application of this estimation, we re-prove the fact that van der Corput and Kronecker sequences are low-discrepancy sequences. Consequently, it follows from the finite gap property that these sequences have $$\alpha $$ α -pair correlations for $$0< \alpha < 1$$ 0 < α < 1 .

Remarks on the pair correlation statistic of Kronecker sequences and lattice point counting

2023

Beitr Algebra Geom

In this short note, we reformulate the task of calculating the pair correlation statistics of a Kronecker sequence as a lattice point counting problem. This can be done analogously to the lattice based approach which was used to (re-)prove the famous three gap property for Kronecker sequences. We show that recently developed lattice point counting techniques can then be applied to derive that a certain class of Kronecker sequences have $$\beta $$ β -pair correlations for all $$0< \beta < 1$$ 0 < β < 1 .