Generation of measures on the torus with good sequences of integers

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Generation of measures on the torus with good sequences

of integers

E. Lesigne, A. Quas, J. Rosenblatt, M. Wierdl

November 14, 2023

Let S:= (s1<s2<. . . )be a strictly increasing sequence of positive inte-

gers and denote e(β):=e2πiβ. We say Sis good if for every real αthe limit

limN1

N∑n≤Ne(snα)exists. By the Riesz representation theorem, a se-

quence Sis good iﬀ for every real αthe sequence (snα)possesses an asymp-

totic distribution modulo 1. Another characterization of a good sequence

follows from the spectral theorem: the sequence Sis good iﬀ in any proba-

bility measure preserving system (X,m,T)the limit limN1

N∑n≤Nf(Tsnx)

exists in L2-norm for f∈L2(X).

Of these three characterization of a good set, the one about limit mea-

sures is the most suitable for us, and we are interested in ﬁnding out what

the limit measure µS,α:=limN1

N∑n≤Nδsnαon the torus can be. In this

ﬁrst paper on the subject, we investigate the case of a single irrational α. We

show that if Sis a good set then for every irrational αthe limit measure µS,α

must be a continuous Borel probability measure. Using random methods,

we show that the limit measure µS,αcan be any measure which is absolutely

continuous with respect to the Haar-Lebesgue probability measure on the

torus. On the other hand, if νis the uniform probability measure supported

on the Cantor set, there are some irrational αso that for no good sequence

Scan we have the limit measure µS,αequal ν. We leave open the question

whether for any continuous Borel probability measure νon the torus there

is an irrational αand a good sequence Sso that µS,α=ν.

Contents

1Introduction, main results 2

1.1 Good sequences, main question 2

1.2 Main results 4

1.3 Weighted averages 8

1.4 Applications in ergodic theory 10

1.5 Future work 10

1.6 Summary of notation 11

2Basic example for representation 11

3Proof of theorem 7.1 for indicators 14

4Measures that cannot be represented at every irrational α23

4.1 Proof of theorem 5.3 24

4.2 Proof of theorem 6.1 24

arXiv:2210.02233v2 [math.CA] 12 Nov 2023

GENERATION OF MEASURES ON THE TORUS WITH GOOD SEQUENCES OF INTEGERS 2

5Representing by weights 25

6Proof of theorem 7.1 for bounded ρ27

6.1 Notes to lemma 28.1 31

7Absolute continuity and positive mean 31

7.1 Proof of theorem 7.2 (b) 31

7.2 Proof of theorem 9.3 (b) 32

8Proof of theorem 7.1 for unbounded ρ33

9The limit measure at rational points 38

10 Examples 39

10.1 Two good sets, but their intersection has no mean. 39

10.2 R1∪R2and R1∩R2have means but are not good 40

10.3 Open set Uwith visit set {n:nα∈U}not good 41

References 43

1 Introduction, main results

Throughout the paper we will use the arithmetic average operator A: for a

ﬁnite index set S, a vector space Vand a S→Vfunction fwe deﬁne

ASf(s)

ASf(s) = As∈Sf(s):=1

#S∑

s∈S

f(s)(2.1)

where #Sdenotes the number of elements in S.

We use the convention that if an interval appears as an index set in

a summation then we consider only the integers in the interval. For

example, ∑n∈[0,N)an=∑n∈{0,1,...,N−1}an.

We also use Weyl’s notation e(β):=e2πiβ. Note that ep(β) = e(pβ)

for every integer p.

We denote by Tthe torus R/Zand we represent it as the unit closed

interval [0, 1]with 0=1.

1.1 Good sequences, main question

2.1 Deﬁnition ▶Good sequence

We say that a sequence S= (sn)n∈Nof integers is good if the limit

limNAn∈[1,N]e(snα)exists for every real number α.

Good sequences have been extensively studied in many parts of mathe-

matics, such as in number theory and ergodic theory.

GENERATION OF MEASURES ON THE TORUS WITH GOOD SEQUENCES OF INTEGERS 3

In this paper we restrict our attention to strictly increasing sequences S

of positive integers in which case we can and will consider Sas a subset of

N, and we’ll use the concept of good sequence and good set interchange-

ably.

Among the wellknown good sequences are the full set Nof positive

integers1, the sequence (n2)n∈Nof squares2and the sequence (pn)n∈Nof 1Weyl 1916.

2Weyl 1916.

primes3where pndenotes the nth prime number. For these sequences the

3Vinogradow 1937.

limits limNAn∈[1,N]e(snα)are as follows

lim

An∈[0,N)e(nα) = (1if α=1

0if α̸=0

lim

An∈[0,N)e(n2α) = 









Ab∈[1,q]eb2a

qif α=a

q,gcd(a,q) = 1

0if αis irrational

lim

An∈[1,N]e(pnα) = 









Ab∈[1,q]

gcd(b,q)=1

e(b/q)if α=a

q,gcd(a,q) = 1

0if αis irrational

(3.1)

In case of a good sequence S= (sn)and a ﬁxed α, the existence of

limNAn∈[1,N]e(snpα)for every p∈Zimplies, by uniform approxima-

tion of a continuous T→Cfunction by trigonometric polynomials, that

for every continuous T→Cfunction ϕthe limit limNAn∈[1,N]ϕ(snα)

exists. By the Riesz representation theorem, this implies that the weak

limit limNAn∈[1,N]δsnαof discrete measures An∈[1,N]δsnαon Texists.

By this argument, the existence of limNAn∈[1,N]e(snα)for every α

implies the existence of the limit measure limNAn∈[1,N]δsnαfor every α.

Denote the Haar-Lebesgue probability measure on the torus Tby λand

recall that the Fourier coeﬃcients λ(ep)of λsatisfy

λ(ep) = (1for p=0

0for p∈Z,p̸=0(3.2)

where for a given measure νand ν-integrable function ϕ, we use4the 4and will use troughout the paper

functional notation ν(ϕ)for the integral of ϕwith respect to ν,

ν(ϕ) = Zϕdν(3.3)

GENERATION OF MEASURES ON THE TORUS WITH GOOD SEQUENCES OF INTEGERS 4

For our three good sets the limit measures are as follows.

lim

An∈[1,N]δnα=(Ab∈[1,q]δb/qif α=a/q,gcd(a,q) = 1

λif αis irrational

lim

An∈[1,N]δn2α=





Ab∈[1,q]δb2a

qif α=a

q,gcd(a,q) = 1

λif αis irrational

lim

An∈[1,N]δpnα=









Ab∈[1,q]

gcd(b,q)=1

δb/qif α=a

q,gcd(a,q) = 1

λif αis irrational

(4.1)

What we see in these three examples is that in case of irrational αthe

limit measure is the Haar-Lebesgue measure λand in case of rational

α=a/q,gcd(a,q) = 1, the limit measure is supported on a subset of the

qth roots of unity and appears to be quite uniform on its support. In case

of irrational α, the simplest question is if it’s possible that the limit measure

is not λ. In case of rational α, we can ask if the limit measure always has

to show some kind of uniformity.

Let us consider a good sequence S= (sn). The existence of the limit

limNAn∈[1,N]e(snα)for every αimplies that the weak limit limNAn∈[1,N]δsnα

of discrete measures An∈[1,N]δsnαon Texists for every α. Let us denote

this weak limit measure by µS,α,

µS,α:=lim

An∈[1,N]δsnα(4.2)

The main question we want to investigate in this paper is

4.1 Question ▶Main question

What can the limit measure µS,αbe? Can it be any Borel probability

measure on T?

1.2 Main results

As we stated earlier, we try to answer question 4.1 for strictly increasing

sequences, and unless we say otherwise, we assume from now on that

S= (sn)is a strictly increasing sequence of positive integers which we

often consider as a subset of N.

Our ﬁrst observation is that the answer to question 4.1 will depend on

α. If αis a rational number, say, α=a

qwith gcd(a,q) = 1, then the limit

measure is clearly supported on the set

Tq:={b/q:b∈[1, q]}(4.3)

of qth roots of unity. So the question is if the limit measure µS,a/qcan

be any probability measure supported on Tq? The answer is yes. First a

terminology.

GENERATION OF MEASURES ON THE TORUS WITH GOOD SEQUENCES OF INTEGERS 5

5.1 Deﬁnition ▶Representable measure at α

Let Sbe a good set, and let νbe a nonzero, ﬁnite Borel measure on T.

We say that Srepresents νat α∈Tif µS,α=1

ν(T)ν.

We say νis representable at αif there is a good set which represents νat

α.

5.2 Theorem ▶Every probability measure on Tqcan be represented

Let qand abe positive integers with gcd(a,q) = 1, and let νbe a

probability measure supported on the set Tqof qth roots of unity.

Then νcan be represented at a

q, that is, there is a good set Sso that

µS,a

q=ν.

Before discussing the limit measure µS,αfor irrational α, let us note the

following fact which will help us appreciate the concept of a good set.

Suppose we are given an irrational number α∈Tand a Borel prob-

ability measure νon T. We claim that there exists a sequence (xn)in T

with asymptotic distribution ν, i. e. such that limNAn∈[1,N]δxn=ν.

Considering such a sequence and using the density of the sequence (nα)n

in T, we can select a strictly increasing sequence (sn)of integers so that

limn(snα−xn) = 0 mod 1, and we have limNAn∈[1,N]δsnα=ν. Taking

S={sn:n∈N}, we could say that µS,α=ν, but nothing insures us

that the set Sis good.

There are diﬀerent ways to prove the preceding claim. For example

we can pick the numbers xnrandomly and independently with law ν, and

the strong law of large numbers asserts that the sequence (xn)has, almost

surely, the right asymptotic distribution.

It is particularly simple to get a point-mass as a limit measure. For

example, to get the Dirac measure at 1/2, so ν=δ1/2, take a strictly

increasing sequence (sn)of natural numbers so that snαconverges to 1/2

mod 1, and let S:={sn:n∈N}. In contrast to this example, for good

sets we have a dramatic departure from the case of rational α.

5.3 Theorem ▶µS,αis continuous for irrational α

Only continuous measures can be represented at an irrational number.

To spell this out, let S= (sn)be a good sequence and αbe an irrational

number.

Then the limit Borel probability measure µS,α=limNAn∈[1,N]δsnαis a

continuous measure.

The obvious question in turn is if any given continuous Borel proba-

bility measure can be represented at any irrational number. The answer is

no, as the next result shows.

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GenerationofmeasuresonthetoruswithgoodsequencesofintegersE.Lesigne,A.Quas,J.Rosenblatt,M.WierdlNovember14,2023LetS:=(s1

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