Poissonian correlations of higher orders

Abstract

We show that any sequence (x n ) n∈N ⊆ [0, 1] that has Poissonian correlations of k-th order is uniformly distributed, also providing a quantitative description of this phenomenon. Additionally, we extend connections between metric correlations and additive energy, already known for pair correlations, to higher orders. Furthermore, we examine how the property of Poissonian k-th correlations is reflected in the asymptotic size of the moments of the function F (t, s, N ) = #{n N : x n − t s/(2N )}, t ∈ [0, 1].

Weak Poissonian Correlations

“…The importance of I β (s, N) in the proofs of our main results is clear from the following lemma, which is an analogue of [9,Proposition 9] in the context of weak correlations.…”

“…The following lemma states that for β < 1, under the assumption of Poissonian (k, β) -correlations, the asymptotic size of R k is the same as that of R * k as N → ∞. In the proof of the lemma we make use of certain facts on R k and R * k that appear in [9]. There are only minor modifications, and these are due to the fact that here we deal with weak correlations; we have chosen to refer the interested reader to the proofs in [9] and explain here briefly where the minor differences come from.…”

“…In the proof of the lemma we make use of certain facts on R k and R * k that appear in [9]. There are only minor modifications, and these are due to the fact that here we deal with weak correlations; we have chosen to refer the interested reader to the proofs in [9] and explain here briefly where the minor differences come from. Lemma 14.…”