WELL-DISTRIBUTION MODULO ONE AND THE PRIMES
J. CHAMPAGNE, T. H. LÊ, Y. R. LIU, AND T. D. WOOLEY
Abstract. Let (pn ) denote the sequence of prime numbers, with 2 = p1 < p2 < . . ..
We demonstrate the existence of an irrational number α having the property that thesequence (αpn ) is not well-distributed modulo 1.

1. Introduction
Consider a real sequence (sn ) and the associated fractional parts {sn } = sn −⌊sn ⌋. Thissequence is said to be equidistributed (or uniformly distributed) modulo 1 when, for eachpair a and b of real numbers with 0 ⩽ a < b ⩽ 1, one hascard{n ∈ [1, N ] ∩ Z : a ⩽ {sn } ⩽ b}
= b − a.
N →∞NA stronger notion than equidistribution is obtained by insisting that, for each natural
number m, the sequence (sn+m ) should be equidistributed modulo 1, uniformly in m.
This property of being well-distributed modulo 1 was introduced by Petersen [5] in 1956.
More concretely, we say that the sequence (sn ) is well-distributed modulo 1 when, foreach pair a and b with 0 ⩽ a < b ⩽ 1, one has
limcard{n ∈ [1, N ] ∩ Z : a ⩽ {sn+m } ⩽ b}
− (b − a) = 0.
N →∞ m∈NNlim sup

It is a consequence of pioneering work of Weyl [9] that, given a polynomialψd (t; α) = αd td + . . . + α1 t + α0 ∈ R[t]with an irrational coeﬃcient αl for some l ⩾ 1, the sequence (ψd (n; α)) is equidistributed
modulo 1. Moreover, Lawton [3, Theorem 2] established that, under the same conditions,this sequence satisﬁes the stronger property of being well-distributed modulo 1. We refer
the reader to Bergelson and Moreira [2, §3] for further discussion on sequences welldistributed modulo 1.
In this note, we focus on the sequence (pn ) of prime numbers with 2 = p1 < p2 < . . ..
It was famously proved by Vinogradov that when α is irrational, then the sequence (αpn )is equidistributed modulo 1 (see [8], for example). Subsequently, a relatively simple proofof this conclusion was presented by Vaughan [7]. It is natural to enquire whether thisequidistribution extends to a corresponding well-distribution property. The purpose ofthis note is to answer this question in the negative.
2020 Mathematics Subject Classiﬁcation. 11K36, 11K06.
Key words and phrases. Well-distribution, prime numbers, equidistribution.
The second author is supported by NSF grant DMS-2246921 and a travel award from the Simons
Foundation. The third author is supported by an NSERC discovery grant. The fourth author is supportedby NSF grants DMS-1854398 and DMS-2001549.
12

J. CHAMPAGNE, T. H. LÊ, Y.-R. LIU, AND T. D. WOOLEY

Theorem 1.1. There exists an irrational number α having the property that the sequence(αpn ) is not well-distributed modulo 1.
At ﬁrst sight, this conclusion may seem surprising, since the primes are undeniably
equidistributed at large scales. However, as ﬁrst discerned by Maier [4], the primes exhibit
irregularities in their distribution at very small scales. When it comes to generating afailure of well-distribution, the most convenient manifestation of such irregularities ofwhich to avail oneself is that established in work of Shiu [6, Theorem 1]. The latter
author shows, in particular, that for each natural number q, there exist arbitrarily long
strings of prime numbers pn+1 , . . . , pn+k with pn+1 ≡ . . . ≡ pn+k ≡ 1 (mod q). By carefully
constructing an associated irrational number α, this congruential bias amongst consecutive
primes may be shown to generate a corresponding failure of well-distribution modulo 1 inthe sequence (αpn ). It will be evident from our proof of Theorem 1.1, which we presentin §2, that many such numbers α can be constructed, each of which is transcendental.
As is usual, we write e(z) for e2πiz , and for θ ∈ R we deﬁne ∥θ∥ = min{|θ − t| : t ∈ Z}.
2. The application of Shiu’s theorem
Our strategy for proving Theorem 1.1 depends on the careful selection of a sequence(nk ) of natural numbers with 1 ⩽ n0 < n1 < . . ., and the associated real numberα=∞X2−nk .

(2.1)

k=0

The sequence (nk ) is deﬁned iteratively in terms of a consequence of Shiu’s theorem on
strings of congruent primes. Thus, for each n ∈ N, there exists an integer m = m(n) withpm+1 ≡ . . . ≡ pm+n ≡ 1 (mod 2n ).

(2.2)

The conclusion of [6, Theorem 1(i)] shows that, when n is suﬃciently large, such an integerm(n) exists with m(n) < exp4 (n), where expr (x) denotes the r-fold iterated exponentialfunction. We deﬁne the sequence (nk ) as follows. We put n0 = 1, and then deﬁne
mk = m(nk ),πk = pmk +nk ,nk+1 = 4πk(k ⩾ 0).

We investigate the well-distribution of the sequence (αpn ) by means of an analogue ofWeyl’s criterion for equidistribution. Thus, as a consequence of [5, Theorems 2 and 3],
we see that (αpn ) is well-distributed modulo 1 if and only if, for each h ∈ N, one haslim sup N

N →∞ m∈N−1N
Xe(hαpn+m ) = 0.

(2.3)

n=1

Before embarking on the proof of Theorem 1.1 in earnest, we pause to conﬁrm that thereal number α deﬁned in (2.1) is irrational.
Lemma 2.1. The number α is transcendental, and hence is irrational.
Proof. For each k ∈ N, putqk = 2nkand ak = 2nkkX
l=02−nl .

WELL-DISTRIBUTION MODULO ONE AND THE PRIMES

3Then we see that ak ∈ N and (qk , ak ) = 1. Moreover, one has
X|qk α − ak | ⩽2nk −lk .
l>k

It is evident from the deﬁnition of the function m(n) via (2.2) that πk > 2nk , whence
nk+1 > 2nk . Consequently, whenever k is large enough and l > k, one hasXX
2nk −nl <2−l−knk < qk−k .
0<l>kl>k

We therefore deduce that |qk α − ak | < qk−k . By Liouville’s theorem (see [1, Theorem 1.1],for example), it therefore follows that α cannot be algebraic. Thus, we conclude that α
is transcendental, and hence irrational.
□We next show that, for each positive integer h, the real number ∥hα(pn −1)∥ is inﬁnitely
often very small for long strings of consecutive primes pn .
Lemma 2.2. Let h be a positive integer. Then for each suﬃciently large positive integerk, one has∥hα(pi+mk − 1)∥ < πk−2 (1 ⩽ i ⩽ nk ).
Proof. Given a positive integer h, when k is suﬃciently large and 1 ⩽ i ⩽ nk , one hasXX
hπk 2−l−3πk < πk−2 .
2−nl ⩽0 < h(pi+mk − 1)l>kl>kMeanwhile, under the same conditions, one ﬁnds that since pi+mk ≡ 1 (mod 2nk ), thenh(pi+mk − 1)kX

2−nl ≡ 0 (mod 1).

l=0

By combining these conclusions, therefore, we infer that for 1 ⩽ i ⩽ nk , one has
XkX−nl−nl2< πk−2 ,h(pi+mk − 1)
2 +l>kl=0□

and the desired conclusion follows from (2.1).

We are now equipped to complete the proof of Theorem 1.1. Let h be a natural number.
From Lemma 2.2, we ﬁnd that whenever k is suﬃciently large and 1 ⩽ i ⩽ nk , one has|e(hαpi+mk ) − e(hα)| = |e(hα(pi+mk − 1)) − 1| < πk−1 .
Thus, when N is an integer with 1 ⩽ N ⩽ nk , thenN−1NXe(hαpn+mk ) − N

n=1

NXe(hα) < πk−1 ,n=1

whence1 − πk−1 <−1N−1NX
n=1e(hαpn+mk ) ⩽ 1.

4J. CHAMPAGNE, T. H. LÊ, Y.-R. LIU, AND T. D. WOOLEY

In particular, for each positive integer N , we infer that1−NX1⩽ sup N −1e(hαpn+m ) ⩽ 1.
Nm∈N
n=1

This relation conﬁrms thatlim sup N −1

N →∞ m∈N

NXe(hαpn+m ) = 1,(2.4)

n=1in contradiction with Weyl’s criterion for well-distribution modulo 1 given in (2.3). We
are therefore forced to conclude that (αpn ) is not well-distributed modulo 1. In view ofLemma 2.1, this completes the proof of Theorem 1.1.
Although the case h = 1 of (2.4) suﬃces to prove Theorem 1.1, we gave a more generalargument since it might be useful. We note also that the number α may be modiﬁed
extensively without impairing the validity of our proof. Indeed, given an integer q ⩾ 2and a sequence of positive integers (bk ) not growing too rapidly, the number α deﬁned in(2.1) could be replaced by∞X
β=bk q −nk ,k=0

and still the sequence (βpn ) is not well-distributed modulo 1. Furthermore, the rapidgrowth of the integer nk may be considerably weakened without damaging the crude
bound of Lemma 2.2, and so the Liouville-type properties of α may also be relaxed.
References[1] A. Baker, Transcendental number theory, Cambridge Math. Lib., Cambridge University Press, Cambridge, 2022.
[2] V. Bergelson and J. Moreira, Van der Corput’s diﬀerence theorem: some modern developments,
Indag. Math. 27 (2016), no. 2, 437–479.
[3] B. Lawton, A note on well-distributed sequences, Proc. Amer. Math. Soc. 10 (1959), no. 6, 891–893.
[4] H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985), no. 2, 221–225.
[5] G. M. Petersen, ‘Almost convergence’ and uniformly distributed sequences, Quart. J. Math. Oxford(2) 7 (1956), no. 1, 188–191.
[6] D. K. L. Shiu, Strings of congruent primes, J. London Math. Soc. (2) 61 (2000), no. 2, 359–373.
[7] R. C. Vaughan, On the distribution of αp modulo 1, Mathematika 24 (1977), no. 2, 135–141.
[8] I. M. Vinogradov, Improvement of some theorems in the theory of primes, C. R. (Doklady) Acad.
Sci. URSS (N.S.) 37 (1942), 115–117.
[9] H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352.
JC and Y-RL: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada
Email address: jchampag@uwaterloo.edu, yrliu@uwaterloo.edu
THL: Department of Mathematics, University of Mississippi, 305 Hume Hall, University, MS 38677, USA
Email address: leth@olemiss.edu
TDW: Department of Mathematics, Purdue University, 150 N. University Street, West
Lafayette, IN 47907-2067, USA
Email address: twooley@purdue.edu