1 Introduction
Let
${\mathcal{A}}\subset \mathbb{N}$
be an infinite sequence of natural numbers and let
$A_{N}$
denote the first
$N$
elements of
${\mathcal{A}}$
. For
$\unicode[STIX]{x1D6FC}\in [0,1]$
, we consider the sequence
$\unicode[STIX]{x1D6FC}{\mathcal{A}}$
, taken modulo 1. Recall that the sequence
$\unicode[STIX]{x1D6FC}{\mathcal{A}}$
is equidistributed in
$\mathbb{R}/\mathbb{Z}$
if for every interval
$I\subset \mathbb{R}/\mathbb{Z}$
one has
$$\begin{eqnarray}\lim _{N\rightarrow \infty }{\displaystyle \frac{1}{N}}\mathop{\sum }_{x\in A_{N}}\unicode[STIX]{x1D7D9}_{I}(\unicode[STIX]{x1D6FC}x)=|I|.\end{eqnarray}$$
For many arithmetic sequences
${\mathcal{A}}$
of interest, the sequence
$\unicode[STIX]{x1D6FC}{\mathcal{A}}$
is equidistributed in
$\mathbb{R}/\mathbb{Z}$
for all irrational
$\unicode[STIX]{x1D6FC}$
. This is true for
${\mathcal{A}}=\mathbb{N}$
itself, or more generally the set of
$k\text{th}$
powers for any
$k\in \mathbb{N}$
, and, most pertinently for us, the set of primes.
In this paper we consider a strictly stronger notion of equidistribution. With notation as above, we define the pair correlation function
$$\begin{eqnarray}F({\mathcal{A}},\unicode[STIX]{x1D6FC},s,N):={\displaystyle \frac{1}{N}}\mathop{\sum }_{\substack{ x_{i},x_{j}\in A_{N} \\ x_{i}\neq x_{j}}}\unicode[STIX]{x1D7D9}_{[-s/N,s/N]}(\unicode[STIX]{x1D6FC}(x_{i}-x_{j})),\end{eqnarray}$$
where both the interval
$[-s/N,s/N]$
and the sequence
$\unicode[STIX]{x1D6FC}{\mathcal{A}}$
are considered modulo
$1$
.
Informally,
$F({\mathcal{A}},\unicode[STIX]{x1D6FC},s,N)$
counts the number of pairs
$(\unicode[STIX]{x1D6FC}x_{i},\unicode[STIX]{x1D6FC}x_{j})$
such that the distance
$\unicode[STIX]{x1D6FC}x_{i}-\unicode[STIX]{x1D6FC}x_{j}\text{ mod }1$
is approximately
$s$
times the average gap length of the sequence
$\unicode[STIX]{x1D6FC}A_{N}\text{ mod }1$
. Analysing the behaviour of
$F({\mathcal{A}},\unicode[STIX]{x1D6FC},s,N)$
for a specific
$\unicode[STIX]{x1D6FC}$
can require delicate diophantine information about
$\unicode[STIX]{x1D6FC}$
(see [Reference Heath-Brown8, Reference Rudnick, Sarnak and Zaharescu12]), but one may instead settle for results that hold for almost all
$\unicode[STIX]{x1D6FC}$
.
In the setting of (1), any
${\mathcal{A}}$
satisfies the equidistribution property for almost all
$\unicode[STIX]{x1D6FC}$
(the sharpest results in this direction are due to Baker [Reference Baker3]). However, in the setting of pair correlations, the situation is more subtle. We say that the sequence
${\mathcal{A}}$
is “metric Poissonian” if for almost all
$\unicode[STIX]{x1D6FC}\in [0,1]$
and for all fixed
$s>0$
, we have
$$\begin{eqnarray}F({\mathcal{A}},\unicode[STIX]{x1D6FC},s,N)=2s(1+o_{{\mathcal{A}},\unicode[STIX]{x1D6FC},s}(1))\end{eqnarray}$$
as
$N\rightarrow \infty$
. Notice that if we had picked
$N$
independent and identically distributed random variables
$(X_{n})_{n\in [N]}$
uniformly distributed on
$\mathbb{R}/\mathbb{Z}$
, instead of the sequence
$\unicode[STIX]{x1D6FC}A_{N}\text{ mod }1$
, then as
$N$
tends to infinity the equivalent pair correlation function would tend to
$2s$
with high probability. Therefore (3) may be viewed as some strong indication that
$\unicode[STIX]{x1D6FC}{\mathcal{A}}\text{ mod }1$
exhibits the behaviour of a random sequence. The connection to the equidistribution property (1) was recently made rigorous: indeed, three simultaneous papers [Reference Aistleitner, Lachmann and Pausinger1, Reference Grepstad and Larcher5, Reference Steinerberger14] recently showed that if (3) holds, for some fixed
$\unicode[STIX]{x1D6FC}$
and for all
$s$
, then for the same
$\unicode[STIX]{x1D6FC}$
one has that
$\unicode[STIX]{x1D6FC}{\mathcal{A}}$
is equidistributed in
$\mathbb{R}/\mathbb{Z}$
.
One might expect that, for the classical sequences
${\mathcal{A}}$
where
$\unicode[STIX]{x1D6FC}{\mathcal{A}}$
is equidistributed for irrational
$\unicode[STIX]{x1D6FC}$
, one could prove that these sequences
${\mathcal{A}}$
are metric Poissonian. Indeed, for
$k\geqslant 2$
and
${\mathcal{A}}$
the set of
$k\text{th}$
powers this was shown by Rudnick and Sarnak [Reference Rudnick and Sarnak11]. However, the sequence
${\mathcal{A}}=\mathbb{N}$
is not metric Poissonian. This follows from consideration of the continued fraction expansion of
$\unicode[STIX]{x1D6FC}$
, but is in fact a special case of a more general phenomenon, connected to the large additive energy of this particular set
${\mathcal{A}}$
.
For a finite set
$B\subset \mathbb{N}$
we define the additive energy
$E(B)$
to be the number of quadruples
$(b_{1},b_{2},b_{3},b_{4})\in B^{4}$
such that
$b_{1}+b_{2}=b_{3}+b_{4}$
. If
$|B|=N$
, then we have the trivial bounds
$N^{2}\ll E(B)\leqslant N^{3}$
. For
$x\in \mathbb{R}$
, let us write
$\Vert x\Vert$
for
$\operatorname{min}_{y\in \mathbb{Z}}|x-y|$
. Then, for an additive quadruple
$(b_{1},b_{2},b_{3},b_{4})$
satisfying
$b_{1}+b_{2}=b_{3}+b_{4}$
, obviously if
$\Vert \unicode[STIX]{x1D6FC}(b_{1}-b_{3})\Vert \leqslant s/N$
then
$\Vert \unicode[STIX]{x1D6FC}(b_{2}-b_{4})\Vert \leqslant s/N$
. This is extremely different behaviour from that which would be seen if
$\unicode[STIX]{x1D6FC}b_{1}$
,
$\unicode[STIX]{x1D6FC}b_{2}$
,
$\unicode[STIX]{x1D6FC}b_{3}$
,
$\unicode[STIX]{x1D6FC}b_{4}$
were genuinely independent and identically distributed uniform random variables on
$\mathbb{R}/\mathbb{Z}$
, and indeed we have the following result of Bourgain, which shows that all sets of nearly maximal energy fail to have the metric Poissonian property.
Theorem 1 [Reference Aistleitner, Larcher and Lewko2, Appendix].
Suppose
$$\begin{eqnarray}\operatorname{limsup}_{N\rightarrow \infty }{\displaystyle \frac{E(A_{N})}{N^{3}}}>0.\end{eqnarray}$$
Then
${\mathcal{A}}$
is not metric Poissonian.
It is clear that the sequence
${\mathcal{A}}=\mathbb{N}$
satisfies the hypotheses of this theorem and therefore this sequence is not metric Poissonian.
Remarkably, a near-converse to this theorem has also been proved to be true.
Theorem 2. Let
$\unicode[STIX]{x1D6FF}>0$
be fixed and suppose that
$E(A_{N})\ll _{\unicode[STIX]{x1D6FF}}N^{3-\unicode[STIX]{x1D6FF}}$
for this fixed
$\unicode[STIX]{x1D6FF}$
and for every
$N$
. Then
${\mathcal{A}}$
is metric Poissonian.
This theorem first appears as statedFootnote
1
in the recent work of Aistleitner et al [Reference Aistleitner, Larcher and Lewko2]. It immediately implies the theorem of Rudnick–Sarnak on
$k\text{th}$
powers and also earlier work on lacunary sequences [Reference Rudnick and Zaharescu13].
It is natural to wonder whether there is a tight energy threshold for this problem. Although the truth seems unlikely to be so clean, it is certainly interesting to consider the behaviour of specific sets
${\mathcal{A}}$
which satisfy
$N^{3-\unicode[STIX]{x1D700}}\ll _{\unicode[STIX]{x1D700}}E(A_{N})\ll o(N^{3})$
for all
$\unicode[STIX]{x1D700}>0$
. In this paper, we prove the following result (answering a question posed by NairFootnote
2
).
Theorem 3 (Main Theorem).
The primes are not metric Poissonian.
When
${\mathcal{A}}$
is the set of primes one has
$E(A_{N})\asymp N^{3}(\log N)^{-1}$
, so certainly the primes are not included in the range of applicability of either Theorem 1 or Theorem 2.
In [Reference Aistleitner, Larcher and Lewko2], Bourgain constructs a sequence
${\mathcal{A}}$
which is not metric Poissonian but nonetheless has
$E(A_{N})=o(N^{3})$
, thereby showing that the converse to Theorem 1 is false. A quantitative analysis of his argument shows that
$E(A_{N})\ll _{\unicode[STIX]{x1D700}}N^{3}(\log \log N)^{-(1/4)+\unicode[STIX]{x1D700}}$
is achievable, for any
$\unicode[STIX]{x1D700}>0$
. So, as an immediate corollary to Theorem 3, we have an improved bound for the smallest energy
$E(A_{N})$
of the initial segments of a set
${\mathcal{A}}$
which is not metric Poissonian.
For further extensions and refinements of Theorems 1 and 2, we refer the interested reader to two very recent preprints [Reference Bloom, Chow, Gafni and Walker4, Reference Lachmann and Technau10].
2 Proof of Theorem 3
The plan of the proof is as follows. For each fixed
$\unicode[STIX]{x1D6FC}$
, we will try to find infinitely many
$n$
such that
$\Vert \unicode[STIX]{x1D6FC}n\Vert$
is extremely small. Using such an
$n$
we will be able to construct a scale
$N$
and a small constant
$s$
such that
$\Vert \unicode[STIX]{x1D6FC}mn\Vert \leqslant s/N$
for some initial segment of integers
$m$
. By a variant of a well-known result concerning the exceptional set for the Goldbach problem, we may show that many such
$mn$
are represented many times as
$p_{i}-p_{j}$
for two primes
$p_{i},p_{j}\leqslant p_{N}$
, the
$N\text{th}$
prime. Combining all these observations will enable us to conclude, provided
$s$
is small enough, that
$F({\mathcal{P}},\unicode[STIX]{x1D6FC},s,N)\geqslant c$
for some constant
$c>2s$
. Since this holds for infinitely many
$N$
, we cannot have
$F({\mathcal{P}},\unicode[STIX]{x1D6FC},s,N)=2s(1+o_{\unicode[STIX]{x1D6FC},s}(1))$
for all almost all
$\unicode[STIX]{x1D6FC}$
and for all
$s>0$
. In fact, we will show that, for almost all
$\unicode[STIX]{x1D6FC}$
, this asymptotic fails to hold.
We now begin to consider the details of this argument. We will use the following result of Harman on diophantine approximation (see [Reference Harman7, Theorem 4.2] and [Reference Harman6]).
Theorem 4. Let
$\unicode[STIX]{x1D713}(n)$
be a non-increasing function with
$0<\unicode[STIX]{x1D713}(n)\leqslant 1/2$
. Suppose that
$$\begin{eqnarray}\mathop{\sum }_{n}\unicode[STIX]{x1D713}(n)=\infty .\end{eqnarray}$$
Let
${\mathcal{B}}$
be an infinite set of integers and let
$S({\mathcal{B}},\unicode[STIX]{x1D6FC},N)$
denote the number of
$n\leqslant N$
,
$n\in {\mathcal{B}}$
, such that
$\Vert n\unicode[STIX]{x1D6FC}\Vert <\unicode[STIX]{x1D713}(n)$
. Then for almost all
$\unicode[STIX]{x1D6FC}$
we have
$$\begin{eqnarray}S({\mathcal{B}},\unicode[STIX]{x1D6FC},N)=2\unicode[STIX]{x1D6F9}(N,{\mathcal{B}})+O_{\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D6F9}(N)^{1/2}(\log \unicode[STIX]{x1D6F9}(N))^{2+\unicode[STIX]{x1D700}})\end{eqnarray}$$
for all
$\unicode[STIX]{x1D700}>0$
, with implied constant uniform in
$\unicode[STIX]{x1D6FC}$
, where
$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(N)=\mathop{\sum }_{n\leqslant N}\unicode[STIX]{x1D713}(n)\end{eqnarray}$$
and
$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(N,{\mathcal{B}})=\mathop{\sum }_{\substack{ n\leqslant N \\ n\in {\mathcal{B}}}}\unicode[STIX]{x1D713}(n).\end{eqnarray}$$
This theorem may be thought of as a flexible version of Khintchines’s theorem on diophantine approximation, in which one can further pass to approximations coming from a set
${\mathcal{B}}$
, provided
${\mathcal{B}}$
is relatively dense. The quality of the error term in this theorem is much better than we need in our application, although it is important that there is no dependence on
$N$
except through
$\unicode[STIX]{x1D6F9}(N)$
. Earlier results of this type include a
$\log N$
factor in the error, which would not have been adequate.
The other technical tool will be the standard bound on the size of the exceptional set in a Goldbach-like problem.
Theorem 5. For a large quantity
$X$
and natural number
$n\leqslant X$
, define
$$\begin{eqnarray}r(n):=\mathop{\sum }_{\substack{ p_{i},p_{j}\leqslant X \\ p_{i}-p_{j}=n}}\log p_{i}\log p_{j}.\end{eqnarray}$$
Then for any
$B>0$
and for all but
$O_{B}(X/\log ^{B}\,X)$
exceptional values of
$n\leqslant X$
, we have the approximation
$$\begin{eqnarray}r(n)=\mathfrak{S}(n)J(n)+O_{B}\bigg({\displaystyle \frac{X}{\log ^{B}X}}\bigg),\end{eqnarray}$$
where
$$\begin{eqnarray}\mathfrak{S}(n):=\left\{\begin{array}{@{}ll@{}}\displaystyle 2\mathop{\prod }_{p\geqslant 3}\bigg(1-{\displaystyle \frac{1}{(p-1)^{2}}}\bigg)\mathop{\prod }_{\substack{ p|n \\ p\geqslant 3}}{\displaystyle \frac{p-1}{p-2}}\quad & n\text{ even, }\\ 0\quad & n\text{ odd}\end{array}\right.\end{eqnarray}$$
is the singular series and
$$\begin{eqnarray}J(n)=\int _{-\infty }^{\infty }\bigg|\int _{0}^{X}e(\unicode[STIX]{x1D6FD}u)\,du\bigg|^{2}e(-\unicode[STIX]{x1D6FD}n)\,d\unicode[STIX]{x1D6FD}\end{eqnarray}$$
is the singular integral.
Proof. This result follows by trivial modifications of the usual argument for the binary Goldbach problem, originally due (independently) to van der Corput, Estermann and C̆udakov. The clearest reference is [Reference Vaughan15, Ch. 3.2] or, for a more modern approach, one may consider the proof of Theorem 19.1 in [Reference Iwaniec and Kowalski9, Ch. 19].◻
We combine these two key ingredients in the following proposition.
Proposition 6. There exists a small absolute
$c>0$
such that, for almost all
$\unicode[STIX]{x1D6FC}\in [0,1]$
, and for all fixed
$s>0$
there exist infinitely many
$n$
satisfying:
-
(i)
$\Vert \unicode[STIX]{x1D6FC}n\Vert <s/n\log n$
; -
(ii) at least
$c\log n$
of the numbers
$n,2n,\ldots ,\lfloor (\log n)/10\rfloor n$
are expressible in at least
$cn/\log \,n$
ways as the difference
$p_{1}-p_{2}$
of two primes
$p_{1},p_{2}\leqslant (n\log n)/2$
.
Proof of Proposition 6.
Let
$c>0$
be a quantity to be specified later. With this
$c$
, let
${\mathcal{B}}$
be the set of natural numbers
$n$
that satisfy (ii) and let
$\unicode[STIX]{x1D713}(n)=\operatorname{min}(1/2,s/n\log n)$
. It is to this
${\mathcal{B}}$
and this
$\unicode[STIX]{x1D713}$
that we will apply Theorem 4.
We claim that
${\mathcal{B}}$
is relatively dense. Indeed, let
$K$
be a large integer and let
$n$
and
$m$
be natural numbers restricted to the ranges
$K\leqslant n<2K$
and
$1\leqslant m\leqslant \lfloor (\log 2K)/10\rfloor$
. For notational convenience we let
$X$
denote the quantity
$(K\log K)/2$
and we consider Theorem 5 with this
$X$
. We say that the pair
$(n,m)$
is exceptional if
$nm$
lies in the exceptional set from Theorem 5 for which the asymptotic formula (5) fails to hold. (Note that
$nm\leqslant X$
, so Theorem 5 applies in this setting.)
The map
$(n,m)\mapsto nm$
is at most
$(\log 2K)/10$
-to-
$1$
, due to the restricted range of
$m$
. As the exceptional set from Theorem 5 has size at most
$O_{B}(K/\!\log ^{B}K)$
, for all
$B>0$
, we conclude that there are at most
$O_{B}(K/\log ^{B}\,K)$
exceptional pairs
$(n,m)$
, for all
$B>0$
. So certainly there are at least
$(1-O_{B}(\log ^{-B}K))K$
values of
$n\in [K,2K)$
such that the asymptotic formula for
$r(nm)$
holds for all
$m\leqslant (\log 2K)/10$
. Let
$D_{K}$
denote this set of
$n$
.
The set
$D_{K}$
is certainly very dense in
$[K,2K)$
and we claim further that
$D_{K}\subset {\mathcal{B}}$
, provided we choose
$c$
small enough. Combining the different scales
$K$
will allow us to show that
${\mathcal{B}}$
is suitably dense. Indeed, let us analyse the asymptotic formula for
$r(nm)$
. When
$nm$
is even, the singular series
$\mathfrak{S}(nm)$
is always
$\unicode[STIX]{x1D6FA}(1)$
. By Fourier inversion, the singular integral is exactly
$(\unicode[STIX]{x1D7D9}_{[0,X]}\ast \unicode[STIX]{x1D7D9}_{[-X,0]})(nm)$
, which is
$\unicode[STIX]{x1D6FA}(X)$
provided that
$|nm|\leqslant X/5$
. But, by the choice of ranges for
$n$
and
$m$
, this inequality is always satisfied. So, for all
$n\in D_{K}$
and for all
$m\leqslant (\log n)/10$
, such that
$nm$
is even, we have
$r(nm)\gg X$
.
Removing the log weights on the primes and recalling the definition of
$X$
, in particular we notice that there is some small absolute constant
$c_{1}$
such that the following holds: if
$n\in D_{K}$
and if
$nm$
is even, there are at least
$c_{1}K/\!\log K$
pairs of primes
$(p_{i},p_{j})$
with
$p_{i},p_{j}\leqslant (K\log K)/2$
such that
$p_{i}-p_{j}$
=
$nm$
. By the definition of
${\mathcal{B}}$
, provided
$c$
is chosen smaller than
$\operatorname{min}(c_{1},1/100)$
, we have that
$D_{K}\subset {\mathcal{B}}$
.
We may now prove that
$\unicode[STIX]{x1D6F9}({\mathcal{B}},N)\gg _{s}\unicode[STIX]{x1D6F9}(N)$
for large
$N$
. Indeed, for
$k_{0}$
a suitably large constant, depending on
$s$
,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}({\mathcal{B}},N) & = & \displaystyle \mathop{\sum }_{\substack{ n\leqslant N \\ n\in {\mathcal{B}}}}\operatorname{min}\bigg({\displaystyle \frac{1}{2}},{\displaystyle \frac{s}{n\log n}}\bigg)\nonumber\\ \displaystyle & \gg & \displaystyle -O_{s}(1)+\mathop{\sum }_{k=k_{0}}^{\lfloor \log _{2}N\rfloor -1}\mathop{\sum }_{\substack{ 2^{k}\leqslant n<2^{k+1} \\ n\in {\mathcal{B}}}}{\displaystyle \frac{s}{2^{k}\log (2^{k})}}\nonumber\\ \displaystyle & \gg & \displaystyle -O_{s}(1)+\mathop{\sum }_{k=k_{0}}^{\lfloor \log _{2}N\rfloor -1}\mathop{\sum }_{2^{k}\leqslant n<2^{k+1}}(1-O_{B}(k^{-B})){\displaystyle \frac{s}{k2^{k}}}\nonumber\\ \displaystyle & \gg & \displaystyle -O_{s}(1)+\mathop{\sum }_{k=1}^{\lfloor \log N\rfloor }{\displaystyle \frac{s}{k}}\nonumber\\ \displaystyle & \gg _{s} & \displaystyle \log \log N\nonumber\\ \displaystyle & \gg _{s} & \displaystyle \unicode[STIX]{x1D6F9}(N).\nonumber\end{eqnarray}$$
Therefore, applying Theorem 4 to this set
${\mathcal{B}}$
and this function
$\unicode[STIX]{x1D713}$
, the main term from the conclusion of Theorem 4 dominates the error term. We conclude that for almost all
$\unicode[STIX]{x1D6FC}$
there are infinitely many
$n\in {\mathcal{B}}$
satisfying
$\Vert \unicode[STIX]{x1D6FC}n\Vert <s/n\log n$
. The proposition is proved.◻
With this moderately technical proposition proved, the deduction of Theorem 3 is extremely short.
Proof of Theorem 3.
Let
$\unicode[STIX]{x1D6FA}\subset [0,1]$
be the full-measure set of
$\unicode[STIX]{x1D6FC}$
for which Proposition 6 holds. Let
$c$
be the constant from Proposition 6 and fix some
$s$
satisfying
$0<2s<c^{2}$
. Let
$\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6FA}$
and fix a large
$N$
to be one of the infinitely many natural numbers which satisfy the conclusions of Proposition 6.
By construction, we know that
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6FC}N\Vert <{\displaystyle \frac{s}{N\log N}}.\end{eqnarray}$$
Therefore, for all
$d\leqslant (\log N)/10$
, we have
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6FC}dN\Vert <{\displaystyle \frac{s}{N}}.\end{eqnarray}$$
But by the second conclusion of Proposition 6, this implies that there are at least
$c^{2}N$
pairs of distinct primes
$p_{i},p_{j}\leqslant (N\log N)/2$
such that
$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6FC}(p_{i}-p_{j})\Vert <{\displaystyle \frac{s}{N}}.\end{eqnarray}$$
As
$p_{N}\sim N\log N$
and
$N$
is large, this certainly implies that
$$\begin{eqnarray}F({\mathcal{P}},\unicode[STIX]{x1D6FC},s,N)\geqslant c^{2}>2s.\end{eqnarray}$$
This holds for infinitely many
$N$
and, therefore, for all
$\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6FA}$
we have
$$\begin{eqnarray}F({\mathcal{P}},\unicode[STIX]{x1D6FC},2s,N)\neq 2s(1+o(1))\end{eqnarray}$$
as
$N\rightarrow \infty$
. As
$\unicode[STIX]{x1D6FA}$
has measure 1, Theorem 3 is proved.◻
Acknowledgements
The author would like to thank Professor R. Nair for making him aware of the central question of this paper, Professor C. Aistleitner for comments on an earlier version, and Professor B. J. Green for his doctoral supervision. A helpful conversation was also had with Sam Chow. The work was completed while the author was a Program Associate at the Mathematical Sciences Research Institute in Berkeley, which provided excellent working conditions. The author is supported by EPSRC grant no. EP/M50659X/1.
