L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Javier CILLERUELO^†& Jean-Marc DESHOUILLERS Gaps in sumsets ofspseudos-th powers

Tome 67, n^o4 (2017), p. 1725-1738.

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GAPS IN SUMSETS OF s PSEUDO s-TH POWERS

by Javier CILLERUELO^† & Jean-Marc DESHOUILLERS (*)

Abstract. — We study the length of the gaps between consecutive members in the sumsetsAwhenAis a pseudos-th power sequence, withs>2. We show that, almost surely, lim sup(bn+1−bn)/logbn =s^ss!/Γ^s(1/s), wherebn are the elements ofsA.

Résumé. — On étudie la taille des différences entre les termes consécutifs de la suitesAoùAest une suite de pseudo-puissancess-ièmes avecs>2. On montre qu’on a presque sûrement lim sup(bn+1−bn)/logbn=s^ss!/Γ^s(1/s), où lesbnsont les éléments de la suitesA.

1. Introduction

Erdős and Rényi [3] proposed in 1960 a probabilistic model for sequences Agrowing like thes-th powers: they build a probability space (U,T, P) and a sequence of independent random variables (ξn)_n∈N with values in{0,1}

and P(ξn = 1) = ¹_sn^−1+1/s; to any u ∈ U, they associate the sequence of positive integers A = Au such that n ∈ Au if and only if ξn(u) = 1.

In short, the events{n∈A} are independent andP(n∈A) = ¹_sn^−1+1/s. The counting function of these random sequencesAsatisfies almost surely the asymptotic relation|A∩[1, x]| ∼x^1/s, whence the terminologypseudo s-th powers. Erdős and Rényi studied the random variablers(A, n) which counts the number of representations ofn in the form n=a1+· · ·+as,

Keywords:Additive Number Theory, Pseudos-th powers, Probabilistic method.

Math. classification:11B83.

(*) The first author was supported by MINECO project MTM2014-56350-P and ICMAT Severo Ochoa project SEV-2015-0554.

The second author has been partly supported by the Indo-French Centre for the Pro- motion of Advanced Research - CEFIPRA, project No 5401-1.

Both authors are thankful to Ecole Polytechnique which made their collaboration easier.

Javier Cilleruelo untimely passed away on May 15th, 2016. I express my deep sorrow for the loss of a brilliant collaborator and a friend. J-M. D.

a₁ 6 · · · 6 a_s, a_i ∈ A. For the simplest case s = 2 they proved that r2(A, n) converges to a Poisson distribution with parameter π/8, when n→ ∞. They also claimed the analogous result fors >2 but their analysis did not take into account the dependence of some events. J. H. Goguel [4]

proved indeed that for each integerd, the sequence of the integersnsuch that rs(A, n) = d has almost surely the density λ^d_se^−λs/d!, where λs = Γ^s(1/s)/(s^ss!). B. Landreau [5] gave a proof of this result based on corre- lation inequalities and also showed that the sequence of random variables (rs(A, n))n converges in law towards the Poisson distribution with param- eterλ_s.

In particular, both the sets of the integers belonging, or not belonging, tosA={a₁+· · ·+as : ai ∈A}have almost surely a positive density and it makes sense to study the length of the gaps insA. The aim of the paper is to obtain a precise estimate for the maximal length of such gaps.

Theorem 1.1. — For any s > 2 the sequence sA = (b_n)_n, sum of s copies of a pseudos-th power sequenceA, satisfies almost surely

(1.1) lim sup

n→∞

b_n+1−b_n logbn

= s^ss!

Γ^s(1/s).

We remark that this result is heuristically consistent with the fact that for a random sequenceSwithP(n∈S) = 1−e^−λ, we have lim sup(s_m+1− sm)/logsm= 1/λ, an exercise on Borel–Cantelli Lemma.

2. Notation, hint of the proof and general lemmas 2.1. Notation

We retain the notation of the introduction, for the probability space (U,T, P) and the definition of the random sequencesA =Au, where the events {n ∈ A} are independent and P(n ∈ A) = ¹_sn^−1+1/s. We further use the following notation.

(1) We write ω to denote a set of distinct positive integers and we denote byE_ω andE_ω^c the events

Eω={ω⊂A} and E^c_ω={ω6⊂A}

respectively. We writeω∼ω⁰ to mean that ω∩ω⁰6=∅but ω6=ω⁰; we remark thatω ∼ ω⁰ if and only if the events E_ω and E_ω⁰ are distinct and dependent.

Ifω={x1, . . . , x_r} we write

σ(ω) ={a1x1+· · ·+arxr: a1+· · ·+ar=s, ai>1}

for the set of all integers which can be written as a sum ofsintegers using all the integersx₁, . . . , x_r. For an integerz we let

Ωz={ω: z∈σ(ω)}.

(2) Given α > 0, we denote by I_i the interval [i, i+αlogi] and we denote byF_i^(α), or simplyF_i when the context is clear, the event

Fi=F_i^(α)={sA∩Ii=∅}.

We denote by ΩIi the family of sets ΩI_i={ω: σ(ω)∩Ii6=∅}.

(3) We letλs= Γ^s(1/s) s!s^s .

(4) We use Vinogradov’s notation , where f g is equivalent to Landau’s notationf =O(g).

2.2. Hints for the proof to Theorem 1.1

We wish to prove that forα > λ⁻¹_s , the eventF_i^(α), defined in Section 2.1, occurs — almost surely — for only finitely manyi’s and that forα < λ⁻¹_s it occurs — almost surely — for infinitely manyi’s. There is a flavour of Borel–Cantelli and indeed a key point in the proof is Lemma 3.5 which asserts relation (3.2), namely

(2.1) P(F_i^(α)) =i^{−αλs+o(1)}.

Let us first see how we can obtain that relation. Hereαis fixed and we do not mention it anylonger. By the definition, the eventFi occurs if and only if for any family of snon necessarily distinct integers which sum up to an integer inI_i, at least one of them is not inA; with our notation, this leads to

Fi= \

ω∈Ω_Ii

E^c_ω.

If theω’s which are involved had pairwise empty intersections, the events E^c_ωwould be independent and we would have

P(F_i) = Y

ω∈Ω_Ii

P(E_ω^c).

Although this is not the case, the structure of the eventsE_ω, which are finite intersections of events taken from from an independent family, permits us to use Harris’ inequality (or FKG inequality, cf. Theorem 2.2 below) to get the lower bound

P(Fi)> Y

ω∈Ω_Ii

P(E_ω^c).

It also permits us, thanks to Janson’s Correlation Inequality (cf. Theo- rem 2.2 below), to get the upper bound

P(F_i)6 Y

ω∈Ω_Ii

P(E_ω^c)×exp 2 X

ω∼ω⁰

P(E_ω∩E_ω⁰)

! ,

where the notationω∼ω⁰ is defined in Section 2.1. It is then a matter of computation, based on Lemma 2.3, to get the central inequality (2.1).

Whenα > λ⁻¹_s the series P

iP(F_i^(α)) converges and the Borel–Cantelli lemma immediately implies that for suchαthe eventsF_i^(α)almost surely occur for only finitely manyi’s.

When α < λ⁻¹_s the series P

iP(Fi) diverges, but this is not enough to conclude directly since the events F_i’s are not independent. However, P. Erdős and A. Rényi proved that a weak dependence among the Fi’s is sufficient for obtaining an “inverse Borel–Cantelli” result (cf. Theorem 2.1 below). It is thus important to have a small upper bound forP(F_i∩F_j)− P(Fi)P(Fj) in average. With our notation, we have

Fi∩Fj = \

ω∈Ii∪Ij

E_ω^c,

and here again Janson’s inequality will help us to obtain a suitable bound.

2.3. Probabilistic results

We use the following generalization of the Borel–Cantelli Lemma, proved indeed by P. Erdős and A. Rényi in 1959 [2].

Theorem 2.1 (Borel–Cantelli Lemma). — Let(Fi)_i∈Nbe a sequence of events and letZ_n=P

i6nP(F_i).

If the sequence(Zn)n is bounded, then, with probability1, only finitely many of the eventsFi occur.

If the sequence(Z_n)n tends to infinity and

n→∞lim P

16i<j6nP(Fi∩Fj)−P(Fi)P(Fj)

Z_n² = 0,

then, with probability1, infinitely many of the eventsFi occur.

The next result, which combines Harris’ inequality and Janson’s Corre- lation Inequality, can be found in [1].

Theorem 2.2. — Let(Eω)_ω∈Ωbe a finite collection of events which are intersections of elementary independent events and assume thatP(E_ω)6 1/2 for anyω∈Ω. Then

Y

ω∈Ω

P(E_ω^c)6P \

ω∈Ω

E_ω^c 6 Y

ω∈Ω

P(E_ω^c)×exp 2 X

ω∼ω⁰

P(E_ω∩E_ω⁰) ,

whereω∼ω⁰ means that the eventsEωandEω⁰ are dependent events.

2.4. A technical lemma

Lemma 2.3. — Given16t6s−1and positive integers a1, . . . , at we have, asz tends to infinity:

X

16x1,...,xt

a₁x₁+···+atx_t=z

(x1· · ·xt)^−1+1/sz^−1+t/s. (1)

X

16x₁,...,x_t a1x1+···+atxt<z

(x₁· · ·x_t)^−1+1/s z−(a₁x₁+· · ·+a_tx_t)−2t/s

z^−t/slogz.

(2)

X

16x₁<···<xs

x1+···+xs=z

(x₁· · ·x_s)^−1+1/s∼s^sλ_s. (3)

Proof. — (1) We have

X

16x₁,...,x_t a1x1+···+atxt=z

(x₁· · ·x_t)^−1+1/s

= (a1· · ·at)^1−1/s X

16x1,...,xt

a₁x₁+···+a_tx_t=z

(a1x1· · ·atxt)^−1+1/s

6(a1· · ·at)^1−1/s X

16y1,...,yt

y₁+···+yt=z

(y1· · ·yt)^−1+1/s.

Ify₁+· · ·+y_t =z then at least one of them, say y_t, is greater thanz/t and is determined byy1, . . . , y_t−1. Thus,

X

16x1,...,xt

a₁x₁+···+a_tx_t=z

(x1· · ·xt)^−1+1/sz^−1+1/s X

16y1,...,yt−1<z

(y1· · ·y_t−1)^−1+1/s

z^−1+1/s



 X

16y<z

y^−1+1/s





t−1

z^−1+1/s(z^1/s)^t−1 z^−1+t/s.

(2) We have

X

16x₁,...,x_t a1x1+···+atxt<z

(x₁· · ·x_t)^−1+1/s z−(a₁x₁+· · ·+a_tx_t)−2t/s

= X

16m<z

(z−m)^−2t/s X

16x₁,...,x_t a1x1+···+atxt=m

(x₁· · ·x_t)^−1+1/s

( by (1)) X

16m<z

(z−m)^−2t/sm^−1+t/s

X

16m6z/2

(z−m)^−2t/sm^−1+t/s+ X

z/2<m<z

(z−m)^−2t/sm^−1+t/s z^−2t/sz^t/s+z^−1+t/s X

z/2<m<z

(z−m)^−2t/s z^−t/s+z^−1+t/s

z^1−2t/slogz z^−t/slogz.

Remark 2.4. — Except in the case whens = 2 and t = 1, the upper bound in (2) may be replaced byz^−t/s.

(3) It follows from Lemma 3 of [5].

3. Proof of Theorem 1.1 3.1. Combinatorial lemmas

Lemma 3.1. — We have X

ω∈Ωz

P(E_ω)∼λ_s asz→ ∞.

Proof. — We have

(3.1) X

ω∈Ωz

P(Eω) = X

ω∈Ωz

|ω|=s

P(Eω) + X

ω∈Ωz

|ω|6s−1

P(Eω).

The main contribution comes from the first sum.

X

ω∈Ωz

|ω|=s

P(Eω) = 1 s^s

X

16x1<···<xs

x₁+···+xs=z

(x1· · ·xs)^−1+1/s∼λs

asz→ ∞, by Lemma 2.3 (3). For the second sum we have X

ω∈Ωz

|ω|6s−1

P(Eω)6 X

16r6s−1

X

a₁,...,a_r a₁+···+ar=s

X

16x1,...,xr

a₁x₁+···+a_rx_r=z

(x1· · ·xr)^−1+1/s

( Lem. 2.3 (1)) X

16r6s−1

z^rs−1z^−1/s.

Lemma 3.2. — For anyz6z⁰ we have X

ω∼ω⁰ ω∈Ωz, ω⁰∈Ω_z0

P(Eω∩Eω⁰)z^−1/slogz.

Proof. — If ω ∈ Ωz then there exist some r 6 s and some positive integersa₁, . . . , arwitha₁+· · ·+ar=ssuch thata₁x₁+· · ·+arxr=z.

Thus, any pair of setsω∼ω⁰ withω∈Ωz, ω⁰∈Ωz⁰, z6z⁰ is of the form ω={x1, . . . , xt, ut+1, . . . , ur}

ω⁰ ={x1, . . . , xt, vt+1, . . . , vr⁰}

with 16t6r6r⁰6sand positive integersa₁, . . . , a_randb₁, . . . , b_r⁰ with a1x1+· · ·+atxt +at+1ut+1+· · ·+arur=z

b1x1+· · ·+btxt +bt+1vt+1+· · ·+br⁰vr⁰ =z⁰.

Of course ifr=t thenω={x₁, . . . , x_r} andr⁰>t+ 1. Otherwiseω=ω⁰. Similarly, whenr⁰ =t, we haver>t+ 1.

Given positive integersz, z⁰, t, r, r⁰, a₁, . . . , a_r, b₁, . . . , b_r⁰ we estimate the

sum ∗

X

ω∼ω⁰

P(Eω∩Eω⁰) where the symbolP∗

means that the sum is extended to the pairsω∼ω⁰ satisfying the above conditions. We distinguish several cases according to the values ofrandr⁰.

Caser>t+ 1and r⁰>t+ 1. — We have

∗

X

ω∼ω⁰

P(E_ω∩E_ω⁰)6 X

16x₁,...,x_t a1x1+···+atxt<z b1x1+···+btxt<z⁰

(x₁· · ·x_t)^−1+1/s

× X

16ut+1,...,ur, at+1ut+1+···+arur

=z−(a₁x₁+···+a_tx_t)

(ut+1· · ·ur)^−1+1/s

× X

16v_t+1,...,v_r0, bt+1vt+1+···+b_r0v_r0

=z⁰−(b₁x₁+···+b_tx_t)

(vt+1· · ·vr⁰)^−1+1/s.

By Lemma 2.3 (1) we have

∗

X

ω∼ω⁰

P(E_ω∩E_ω⁰) X

x1,...,xt

a₁x₁+···+a_tx_t<z b₁x₁+···+btxt<z⁰

(x₁· · ·x_t)^−1+1/s

×

z−(a₁x₁+· · ·+a_tx_t)^r−t_s ⁻¹

×

z⁰−(b1x1+· · ·+btxt)^{r0 −t}_s −1

.

Using the inequalityAB6A²+B², we get

∗

X

ω∼ω⁰

P(Eω∩Eω⁰)

6 X

x1,...,xt

a1x1+···+atxt<z

(x₁· · ·x_t)^−1+1/s

z−(a₁x₁+· · ·+a_tx_t)2(r−t−s)/s

+ X

x1,...,xt

b₁x₁+···+b_tx_t<z⁰

(x₁· · ·x_t)^−1+1/s

z⁰−(b₁x₁+· · ·+b_tx_t)2(r⁰−t−s)/s

( Lem.2.3 (2))

z^−t/slogzz^−1/slogz.

Caser=t andr⁰>t+ 1. — In this case we have

∗

X

ω∼ω⁰

P(E_ω∩E_ω⁰)

6 X

16x₁,...,x_t a1x1+···+atxt=z b₁x₁+···+btx_t<z⁰

(x₁· · ·x_t)^−1+1/s

× X

16v_t+1,...,v_r0

b_t+1v_t+1+···+b_r0v_r0

=z⁰−(b1x₁+···+btx_t)

(v_t+1· · ·v_r⁰)^−1+1/s

( Lem. 2.3 (1))6 X

16x₁,...,x_t a1x1+···+atxt=z b₁x₁+···+btx_t<z⁰

(x₁· · ·x_t)^−1+1/s

z⁰−(b₁x₁+· · ·+b_tx_t)^{r0 −t}_s −1

6 X

16x₁,...,x_t a1x1+···+atxt=z

(x₁· · ·x_t)^−1+1/s

z^ts−1z^−1/s.

Caser⁰=tandr>t+1. — This case is similar to the previous one.

Lemma 3.3. — Let α >0 and the letIi be the interval[i, i+αlogi].

For anyi6j we have X

ω∼ω⁰ ω∈Ω_Ii, ω⁰∈Ω_Ij

P(E_ω∩E_ω⁰)i^−1/s(logi)²(logj).

Proof. — We have X

ω∼ω⁰ ω∈Ω_Ii, ω⁰∈Ω_Ij

P(E_ω∩E_ω⁰)6 X

z∈Ii, z⁰∈Ij

X

ω∼ω⁰ ω∈Ωz, ω⁰∈Ω_z0

P(E_ω∩E_ω⁰)

( Lem. 3.2 ) X

z∈Ii, z⁰∈Ij

z^−1/slogz

(logi)²(logj)i^−1/s. Lemma 3.4. — We have

Y

ω∈Ω_Ii

P(E_ω^c) =i^{−αλs+o(1)}.

Proof. — We observe that Y

z∈Ii

Y

ω∈Ωz

P(E_ω^c)6 Y

ω∈Ω_Ii

P(E_ω^c)6 Y

ω∈Ω_Ii

|ω|=s

P(E_ω^c) = Y

z∈Ii

Y

ω∈Ωz

|ω|=s

P(E_ω^c).

WritingP(E_ω^c) = 1−P(Eω) and taking logarithms we have

log Y

z∈I_i

Y

ω∈Ωz

P(E_ω^c)

!

=X

z∈I_i

X

ω∈Ωz

log(1−P(Eω))∼ −X

z∈I_i

X

ω∈Ωz

P(Eω) ( Lem. 3.1 ) ∼ −X

z∈I_i

λs∼ −αλslogi.

On the other hand,

log







 Y

z∈Ii

Y

ω∈Ωz

|ω|=s

P(E_ω^c)









=X

z∈Ii

X

ω∈Ωz

|ω|=s

log(1−P(E_ω))∼ −X

z∈Ii

X

ω∈Ωz

|ω|=s

P(E_ω)

=−X

z∈Ii

X

x₁<···<xs

x₁+···+x_s=z

1

s^s(x₁· · ·x_s)^−1+1/s

( Lem. 2.3 (3)) ∼ −λsαlogi.

Lemma 3.5. — We have

(3.2) P(Fi) =i^{−αλs+o(1)}. Proof. — As we noticed it in Section 2.2, we have

F_i= \

ω∈Ω_Ii

E^c_ω.

SinceP(Eω)61/2 for anyω, Theorem 2.2 applies and we have

Y

ω∈Ω_Ii

P(E_ω^c)6P(Fi)6 Y

ω∈Ω_Ii

P(E_ω^c)×exp









2 X

ω∼ω⁰ ω,ω⁰∈Ω_Ii

P(Eω∩Eω⁰)







 .

After Lemma 3.4 we only need to prove X

ω∼ω⁰ ω,ω⁰∈Ω_Ii

P(Eω∩Eω⁰) =o(1).

But it is a consequence of Lemma 3.3 withj =i.

X

ω∼ω⁰ ω, ω⁰∈Ω_Ii

P(E_ω∩E_ω⁰)i^−1/s+o(1).

Lemma 3.6. — Ifi < j andI_i∩I_j =∅ then Y

ω∈Ω_Ii∪Ω_Ij

P(E_ω^c)6P(Fi)P(Fj)(1 +O(j^−1/slogj)).

Proof. — It is clear that Y

ω∈Ω_Ii∪Ω_Ij

P(E_ω^c) =



 Y

ω∈Ω_Ii

P(E_ω^c)







 Y

ω∈Ω_Ij

P(E_ω^c)







 Y

ω∈Ω_Ii∩Ω_Ij

P(E^c_ω)





−1

.

Harris’ inequality, applied to the first two products, gives Y

ω∈Ω_Ii∪Ω_Ij

P(E_ω^c)6P(Fi)P(Fj)



 Y

ω∈Ω_Ii∩Ω_Ij

P(E_ω^c)





−1

.

The logarithm of the last factor is

− X

ω∈Ω_Ii∩Ω_Ij

log(1−P(E_ω))∼ X

ω∈Ω_Ii∩Ω_Ij

P(E_ω)

SinceIi∩Ij =∅, ifω∈ΩI_i∩ΩI_j then |ω|6s−1. Thus, X

ω∈Ω_Ii∩Ω_Ij

P(E_ω)6 X

ω∈Ω_Ij

|ω|6s−1

P(E_ω)

6 X

z∈Ij

X

16r6s−1

X

a1+···+ar=s

X

16x₁<···6x_r a1x1+···+arxr=z

(x₁· · ·x_r)^−1+1/s

( Lem. 2.3 (1)) j^−1/slogj.

Thus,



 Y

ω∈Ω_Ii∩Ω_Ij

P(E_ω^c)





−1

61 +O(j^−1/slogj)

which ends the proof of the Lemma.

3.2. End of the proof

After those Lemmas we are ready to finish the proof of Theorem 1.1.

Ifα >1/λsthen X

i

P(F_i) =X

i

i^{−αλs+o(1)}<∞

and Theorem 2.1 implies that with probability 1 only finitely many events Fi occur. This proves that

lim sup

k→∞

bk+1−bk

logbk 61/λs. Ifα <1/λsthen

Zn =X

i6n

P(Fi) =X

i6n

i^{−αλs+o(1)}=n^{1−αλs+o(1)}→ ∞.

If in addition

(3.3) lim

n→∞

P

16i<j6nP(Fi∩Fj)−P(Fi)P(Fj)

Z_n² = 0,

Theorem 2.1 implies that with probability 1 infinitely many eventsFioccur and

lim sup

k→∞

bk+1−bk

logbk >1/λs.

Note that P(F_i∩F_j) > P(F_i)P(F_j) for all 1 6 i < j 6 n, so the limit (3.3) is not negative.

We next prove (3.3). We observe that Fi∩Fj= \

ω∈Ω_Ii∪Ω_Ij

E^c_ω,

so we can use Janson inequality to get

P(F_i∩F_j)6 Y

ω∈Ω_Ii∪Ω_Ij

P E_ω^c

×exp











2 X

ω∼ω⁰ ω,ω⁰∈Ω_Ii∪Ω_Ij

P(E_ω∩E_ω⁰)









 .

Observe that X

ω∼ω⁰ ω,ω⁰∈Ω_Ii∪Ω_Ij

P(E_ω∩E_ω⁰)6 X

ω∼ω⁰ ω,ω⁰∈Ω_Ii

P(E_ω∩E_ω⁰) + X

ω∼ω⁰ ω,ω⁰∈Ω_Ij

P(E_ω∩E_ω⁰)

+ X

ω∼ω⁰ ω∈Ω_Ii, ω⁰∈Ω_Ij

P(E_ω∩E_ω⁰).

Applying Lemma 3.3 to the three sums we have X

ω∼ω⁰ ω,ω⁰∈Ω_Ii∪Ω_Ij

P(Eω∩Eω⁰)i^−1/s(logi)³+j^−1/s(logj)³+i^−1/s(logi)²(logj),

and so

(3.4) exp











2 X

ω∼ω⁰ ω,ω⁰∈Ω_Ii∪Ω_Ij

P(Eω∩Eω⁰)











61 +O

i^−1/s(logi)²(logj) .

Thus,

(3.5) P(Fi∩Fj)6 Y

ω∈Ω_Ii∪Ω_Ij

P E^c_ω

× 1 +O(i^−1/s(logi)²(logj)) .

Since α < λs, the number β= (1−αλs)/2 is positive. Now we split the sum in (3.3) into three sums:

∆_1n = X

16i<j6n n^β<i<j−αlogj

P(F_i∩F_j)−P(F_i)P(F_j)

∆_2n = X

16i<j6n i6n^β

P(F_i∩F_j)−P(F_i)P(F_j)

∆_3n = X

16i<j6n j−logj6i6j

P(F_i∩F_j)−P(F_i)P(F_j)

(1) Estimate of ∆1n. Since in this case we have I_i ∩I_j = ∅, we can apply Lemma 3.6 to (3.5) to get

Y

ω∈Ω_Ii∪Ω_Ij

P E_ω^c

6P(Fi)P(Fj)(1 +O(j^−1/slogj)).

This inequality and (3.5) gives

P(Fi∩Fj)6P(Fi)P(Fj)× 1 +O(i^−1/s(logi)²(logj)) ,

so

P(F_i∩F_j)−P(F_i)P(F_j)P(F_i)P(F_j)i^−1/s(logi)²(logj) n^−β/s+o(1)P(F_i)P(F_j).

Thus

(3.6) ∆_1nn^−β/s+o(1) X

i,j6n

P(F_i)P(F_j)n^−β/s+o(1)Z_n².

(2) Estimate of ∆2n. In this case we use the crude estimate (3.7) P(Fi∩Fj)−P(Fi)P(Fj)6P(Fj).

We have (3.8) ∆2n6X

j6n

X

i6j^β

P(Fj)6X

j6n

j^βP(Fj)6n^βZn6n^−β+o(1)Z_n², sinceZn =n^{1−αλs+o(1)} =n^2β+o(1).

(3) Estimate of ∆_3n. Again we use (3.7) and we have (3.9) ∆3n 6X

j6n

X

j−αlogj6i6j

P(Fj)6αlognX

j6n

P(Fj)6n^−2β+o(1)Z_n². Finally, using the estimates in (3.6), (3.8) and (3.9) we have

P

16i<j6nP(Fi∩Fj)−P(Fi)P(Fj) Z_n²

n^−β/s+o(1)+n^−β+o(1)+n^−2β+o(1)→0.

This ends the proof of (3.3) and hence that of Theorem 1.1.

BIBLIOGRAPHY

[1] R. Boppona&J. Spencer, “A useful elementary correlation inequality”,J. Comb.

Theory50(1989), no. 2, p. 305-307.

[2] P. Erdős &A. Rényi, “On Cantor’s series with convergentP

1/qn”,Ann. Univ.

Sci. Budap. Rolando Eötvös, Sect. Math.2(1959), p. 93-109.

[3] ——— , “Additive properties of random sequences of positive integers”,Acta Arith.

6(1960), p. 83-110.

[4] J. H. Goguel, “Über Summen von zufälligen Folgen natürlischer Zahlen”,J. Reine Angew. Math.278/279(1975), p. 63-77.

[5] B. Landreau, “Étude probabiliste des sommes des puissances s-ièmes”,Compositio Mathematica99(1995), no. 1, p. 1-31.

Manuscrit reçu le 21 février 2016, révisé le 7 juillet 2016,

accepté le 15 septembre 2016.

Javier CILLERUELO^†

Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas Universidad Autónoma de Madrid 28049, Madrid (Spain)

Jean-Marc DESHOUILLERS Bordeaux INP, CNRS

Institut Mathématique de Bordeaux, UMR 5251 33405 Talence (France)

[email protected]

†