Some recent results related to Poissonian pair correlation
problems

Wolfgang Stockinger
Johannes Kepler University
Institute for Financial Mathematics and Applied Number Theory

23. Mai 2018

Wolfgang Stockinger

23. Mai 20181 / 13

Poissonian Pair Correlations

Let (xn )n∈N be a sequence of real numbers in [0, 1], and let k · k denote thedistance to the nearest integer. For every interval [−s, s], we set
R2 ([−s, s], (xn )n∈N , N) :=1nso# 1 ≤ l 6= m ≤ N : kxl − xm k ≤.
NN

A sequence (xn )n∈N in [0, 1] is said to have Poissonian pair correlations, iffor each s ≥ 0, R2 ([−s, s], (xn )n∈N , N) tends to 2s as N → ∞.

Wolfgang Stockinger

23. Mai 20182 / 13

Basic facts

For random numbers X1 , X2 , . . . chosen uniformly and independently,
R2 ([−s, s], (Xn )n∈N , N) → 2s,with probability tending to 1 as N → ∞.
Metrical theory of ({an α})n∈N , where (an )n∈N is an integer sequence,is well-understood.
Kronecker sequences ({nα})n∈N are not Poissonian for any α.

Wolfgang Stockinger

23. Mai 20183 / 13PPC and uniform distribution modulo 1

Theorem (Grepstad-Larcher, Aistleitner-Lachmann-Pausinger,Steinerberger)
Let (xn )n∈N be a sequence in [0, 1] and assume that for each s ≥ 0 the paircorrelation function R2 ([−s, s], (xn )n∈N , N) tends to 2s as N → ∞, thenthe sequence is uniformly distributed.

Wolfgang Stockinger

23. Mai 20184 / 13

Gap-Theorem

Theorem (Larcher-S.)
Let (xn )n∈N be a sequence in [0, 1) with the following property: There is ans ∈ N, positive real numbers K and γ , and innitely many N such that thepoint set x1 , . . . , xN has a subset with M ≥ γN elements, denoted byxj , . . . , xjM , which are contained in a set of points with cardinality at mostKN having at most s dierent distances between neighbouring sequence
elements, so-called gaps. Then, (xn )n∈N does not have Poissonian paircorrelations.
1

Wolfgang Stockinger

23. Mai 2018

5 / 13

Applications of Gap-Theorem

Sequences having a suciently large intersection with nite gap sequences(e.g., Kronecker sequences ({nα})n∈N , . . . )Corollary
Let (xn )n∈N be a sequence in [0, 1) with the following property: There is aconstant κ > 0, a sequence N1 < N2 , . . . of positive integers and for each(i)
Ni , i ≥ 1, a Kronecker sequence (yn )n∈N such that
(i)

(i)

|{x1 , . . . , xNi } ∩ {y1 , . . . , yNi }| ≥ κNi ,then (xn )n∈N does not have Poissonian pair correlations.
Corollary
If (an )n∈N is quasi-arithmetic of degree d = 1, then there is no α such thatthe pair correlations of ({an α})n∈N are Poissonian.
Wolfgang Stockinger

23. Mai 20186 / 13

Quasi-arithmetic of degreedDenition
Let (an )n∈N be a strictly increasing sequence of positive integers. We callthis sequence quasi-arithmetic of degree d , where d is a positive integer, if
there exist constants C , K > 0 and a strictly increasing sequence (Ni )i≥1 ofpositive integers such that for all i ≥ 1 there is a subset A(i) ⊂ (an )1≤n≤Niwith A(i) ≥ CNi such that A(i) is contained in a d -dimensional arithmetic
progression P (i) of size at most KNi .

Wolfgang Stockinger

23. Mai 20187 / 13PPC and the additive energy

Additive energy of a set of real numbers A is dened to be
1,X

E (A) :=

a+b=c+dwhere the sum is extended over all quadruples (a, b, c, d) ∈ A4 . One has|A|2 ≤ E (A) ≤ |A|3 .

Theorem (Aistleitner-Larcher-Lewko)
Let (an )n∈N be a sequence of distinct integers and suppose that there
exists a xed constant  > 0 such that
E (AN )  N 3− ,N → ∞,where AN denotes the rst N elements of (an )n∈N . Then for almost all α
one hasR2 ([−s, s], ({an α})n∈N , N) → 2s,

N → ∞,for all s ≥ 0.
Wolfgang Stockinger

23. Mai 20188 / 13PPC and the additive energy

Is it possible for an increasing sequence of distinct integers (an )n∈N
which satises E (AN ) = Ω(N 3 ) that the sequence ({an α})n∈N has
Poissonian pair correlations for almost all α?
If (an )n∈N is an increasing sequence of distinct integers, does
E (AN ) = o(N 3 ) imply that the sequence ({an α})n∈N has Poissonian
pair correlations for almost all α?

Wolfgang Stockinger

23. Mai 2018

9 / 13PPC and the additive energy

Theorem (Bourgain)
If E (AN ) = Ω N 3 , where AN denotes the rst N elements of (an )n∈N ,then there exists a subset of [0, 1] of positive measure such that for every
α from this set the pair correlations of ({an α})n∈N are not Poissonian.
Theorem (Bourgain)
There exist sequences of distinct integers (an )n∈N with E (AN ) = o(N 3 ),such that ({an α})n∈N fails to have the metrical Poissonian pair correlation
property.

Wolfgang Stockinger

23. Mai 201810 / 13PPC and the additive energy

Theorem (Lachmann-Technau)
Suppose that (an )n∈N is a strictly increasing sequence of positive integers.
If E (AN ) = Ω(N 3 ), then the exceptional set has full Lebesgue measure.
Theorem (Aichinger-Aistleitner-Larcher)
For a strictly increasing
sequence (an )n∈N of positive integers we have3E (AN ) = Ω N if and only if (an )n∈N is quasi-arithmetic of some degreed.

Wolfgang Stockinger

23. Mai 201811 / 13

Additive energy, quasi-arithmetic sequences and PPC

Theorem (Larcher-S.)
If E (AN ) = Ω(N 3 ), then there is no α such that the pair correlations of({an α})n∈N are Poissonian.

Wolfgang Stockinger

23. Mai 201812 / 13

Higher-Dimensional concept

Theorem (Hinrichs-Larcher-Ullrich)

Let x 1 , x 2 , . . . ∈ [0, 1)d be such that for every s ∈ N, and with(s)RN :=we have

1N#{1 ≤ l 6= m ≤ N| kx l − x m k∞ ≤sN 1/d},(s)lim RN = (2s)d ,N→∞

then x 1 , x 2 , . . . is uniformly distributed in [0, 1)d .

Wolfgang Stockinger

23. Mai 201813 / 13