on sums of sequences of integers. ii

ON SUMS OF SEQUENCES

Acta Math. Hung. 44 (1--2) (1984), 169--179.

OF INTEGERS. II

A. BALOG and A. SARKOZY (Budapest)

1. Throughout this paper, we use the following notation: cl, c2, ..., No, N1, ... denote positive absolute constants. 0~, 02, ... are real numbers such that ]0~l~l for all i. We write eX=exp (x) and e2~i~=e(c0. The distance from e to the nearest

f 1 \

integer is denoted by Ilc~[I so that []e[[ = rain (c~-[~], [~] + 1 - c 0. We put rain [A, + J =

= A. We denote the least prime factor of n by p (n), while the greatest prime factor of n is denoted by P(n).

2. In this series, we study the arithmetic nature of the numbers of the form a + b where a, b are taken from "dense" sequences of integers. In Part I (see [1]) we showed the existence of a, b such that P(a+b) is "small" (in terms of a+b). Here our goal is first to generalize the results of Part I. In fact, Sections 3--6 will be devoted to the proof of the following theorem:

THEOREM 1. Let N be an integer satisfying N>No, y a real number satisfying

(1) 1 -<_ y _-< J/N,

and let d , ~, vY" be sequences oJ" integers such that (i)

(2) N/2 < a<= N for aE d ,

(3) N/2 < b <= N for bE~ and (4) 2N/y < k <= 4N/y for k EX.

(ii) Writing A = Z 1 , B = Z 1 and K = • 1 ,

aE,~ bE~ kE,:C we have

(5) ABKy 2 >- 106N 3 log N. (iii) I f kE S then

(7)

integers then there exist a, b such that a E d , bE~ and P(a+b) is "large" (in terms

Then there exist integers a, b, k such that aE d , bE~, kE • and

kl(a + b).

In Section 7, we derive from this theorem that if ~r ~ are "dense" sequences of

Acta Mathcma~ca Hungarica 44, 1984

170 A. BALOO AND A. SkRKOZY

of a+b). Finally, we show in Section 8 that the lower bound for P(a+b) obtained in this way can be improved by using a direct method, based on the large sieve.

(8)

_ ( k N "l :t/2 3. eut e=t j

Furthermore, put

O <

so that in view of (1) and (4), for large N we have

- - N < ( 4 N ~ ]a/~ [ ~ J = I, l o g N ) < N.

S~ (~) = Z d. e (n~) n ~ x

2N

s = SOl = Z d . ,

2N+U--1

S(~)U(a) = Z v .e(n~)

F(~) = ,~ e(a~), aE,~r

U = [8N/y]+ l, d. = Z 1, mk=n m<--y kE~ff

2N (for 0 <- x <= 2N), S(~) = S2N(~) = Z d . e ( n ~ ) ,

n = l

U--1

U(a) = ~ e(n~), tl=O

(so that v . = .~ n - - U < j ~ n

6(~) = Z e(b~) b E ~

and (in view of (2) and (3))

H(~) = F(a)G(~)= .~ e ( (a+b)a)= a E .~ b E ~

(so that h , = • 1). a+b=n a E .~ b E ~

We start out from the integral

1

J = f F(~z)G(~z)S(-~)d~ -- 0

~N

h . e ( n ~ ) n = N + l

1 1 2N 2N 2N

= fH(~)S( -~)d~= f Z Zh.dme((n-m)~) d~= Z h.d.. 0 0 n = N + I m = l n=N-{-I

Obviously, d , > 0 implies the existence of an integer k such that kin, while h,>O implies that n can be written in the form a+b=n (aE~r bE~). Thus in order to prove the solvability of (7), it is sufficient to show that

2N

(9) ] = Z h . a . > O . n = N + I

In order to prove this (by using the Hardy--Littlewood method), we need some lem- n l a s .

Ac ta Mathernat ica Hungar~ca 44. 1984

ON SUMS OF SEQUENCES OF INTEGERS. II 171

4. In this section, we assert some preliminary lemmas.

LEMgA 1. I f L is a positive integer, c~ a real number then we have

ie(ncQ- L < 4L~[c~l.

This is identical with Lemma 2 in [1].

LEMMA 2. For arbitrary real numbers ~, M we have

<- min M, 1=<m~=<=M 2['~-~ "

See e.g. [3], p. 9.

LEMMA 3. I f c~, V are real numbers and a, q, fare integers such that q > O, (a, q) = 1

ql 1 and c~- <=--~ then we have

s'+q [ l min IV, J ~ 6V+ q log q.

re=y+1 , 2llc~mll -

See e.g. [3], p. 23.

LEMMA 4. I f C~, M, V are real numbers and a, q are integers such that M>-I, a 1

q > O, (a, q) = l and ~ - - ~ <=-~7 then we have

m=~ 2[I~mll = +1 (6V+qlogq) = 6 M V + 6 v + ( M + q ) l o g q . q

This is identical with Lemma 5 in [1].

5. In this section, we estimate S, S(~) and v,.

LEMMA 5. We have S<=Ky.

PROOF. 2N 2N

S=- Z d , = X X 1= 2 X n = l n = l mk=n m<=y k<=N/m

m<=y kE~ff k E ~

1 ~ X X I = K[yl <: KY. m<=y kE~C

LEMMA 6. I f l <=u<=2N and a, q are integers such that 2 ~q<- N/Q and (a, q ) = l then we have IS,(a/q)l<=Kq.

PROOF. We have q

(10) S,(a/q) ~ d,e(na/q) = Z ( • d,)e(b/q). n ~ u b= l n<--u

na ~ b(mod q)

Ac ta M a th ema t i ca Hungar ica 44~ 198~

172 A. BALOG AND A. SARKOZY

Here the inner sum can be rewritten in the following form:

O1) Z d .= Z Z 1--- n~_u n<--u mk=n

na=--b(modq) na=-b(mod q) m~_y kE ~d

Z I = Z Z 1 = Z Z 1+ mk~u k<=u m<=u/k k<=uly m<=y

mka=--b(modq) kE;~ r mka=--b(modq) kE~d mka=--b(modq) m<=y m~_y kE ~d

~ ' Z 1. ulY<k<=u m~_ulk

k E ;,~ mka~b(mod q)

By (6), we have ( N I _ ~ N ) ~/2 N

p ( k ) > ~ - - = Q = q .

This inequality and (a, q)= 1 imply that (ka, q)= 1 hence

U Z 1 = Y+01 and Z 1 = --~-q +02

m ~ y q m~ulk mka =--b(mod q) mka =-- b(mod q)

(where 01=01(a, b, k), 02=02(a, b, k)). Thus we obtain from (10) and (11) that

q

Su (a/q) = Z ( Z dn) e (b/q) = b = l n~_u

na =--- b(mod q)

[ y+0,/+ I} =l ~k~_uty q , utr<k<:u - - ~ + 03 e(b/q) k E ,.Y" k E ,, ~f

{( y } b~=l k~--u[y q uly<k~_u q

= Z - - + Z ~ + O z k ~ l e ( b / q ) =

kE~ r kE~ r kE~e"

: + e(b/q) + (04 .~ 1) = 05qK <= qK k _ y u]y<k~_u ~ / b = l b = l k E ~ k E s r kEJd

q

(where 03, 04, 05, 06 depend on a, b, k) since by q>-2, ~e(b/q)=O. b = l

LEMMA 7. I f C~ is a real number and a, q are integers such that 2<:q<=N/Q, a a < 1

(a, q)= 1 and - -~ - ~ then we have

[S(a)t < 14 (KUlog N) 1/~.

l _ a < 1 Then PROOF. We write f l=a_qa so that [fl]= by using Lemma q qQ I

6 and the well-known inequality ] 1 - e(fl)] <- 2n l fl[, we obtain by partial summation

A c t a Ma thema t i ca Hungar$ca 44, 1984


that

IS (~ = . =1 (5;. (a/q) - S._ ~ (a/q)) e (nil) =

= ~=~= S.(a/q) (e(nfl) - e((n + Off)) + S.(a/q) e ( (2g+ 1 <=

2N

<= ~ gq[1 - e(fl)[ + Kq <= Kq(2N2n Ifl] + l) <

< Kq 2N2n +1 =4re +Kq~_4n -~ Q <

log N 1/2 = } =

LI?MMa 8. I f ~ & a real number and a, q are integers such that N/Q<q<=Q,

(a, q)= l and ~ - = - ~ then we have

[S(e)I < 3(KNlog N) 1/~.

PROOF. By using Lemmas 2 and 4, and with respect to (1), (4), (5) and (8), we obtain that

IS(~)I = I~l.=ldne(n~)l = Imk~--~Ne(mk~)[= =

|

m~=y kEae"

=IX( Z e(mk~))l~=X.t ZN/~,,)e(mk~)l~-- k m~-min(2Nlk, y) k m~mi

( <-- ,~ min min(2N/k,y), 1 -<- Z, min y, ~_ k~=4N[y k~_4N[y

4N

_ q + 6 y + + q l o g q <

( 4N ) < 24 ~ + 6 ~ + 10~g./~ 0og ~'~/(ABK) ~ ~ Q log Q <

Q(IogN) 1/~ [ 1 f ABK ]112 } < 24Q + 6 Kll2 b I , ] -~ i . ~ ) + Q log N <

< " ] " 0 - 0 1 , ~ ) +2 I,l~gN] } I ~ 1 7 6 2001(AB)l/2N F 1) -<

1,~( 1 (N.N) 11~ ) < 2 ( g Y l o g N ) ' (2{)0 N ~-l < 3(KNlogg ) 1/2.

A c t a M a t h e m a t i c a Hungar i ca 44, I984

174 A. BALOG AND A. SA.RKOZY

LEMMA 9. I f

1 1 (12) --<=< 1---

O Q then we have (13) IS(~)I < 14 (KNlog N) v~.

PROOF. By Dirichlet's theorem, there exist integers a, q such that l<=q<=Q,

~a~,=~ aod I ~ 1 ~ ~ (=~/ 70 . (12) implies that q > l . If 2<=q<=N/Q then (13) is a consequence of Lemma

7while if N/Q<q<=Q then (13) holds by Lemma 8.

LEMMA 10. I f n is a positive integer satisfying U<-n<=2N then we have v.>=K.

PROOF. For U<=n~=2N we have

v.= ~ 4 = ~ 2 1 . = 2 1 = 2 Z 1= j = n - - U + l j = n - - U + I mk= j n - - U < m k ' ~ n kE :#" n--U n

<rrl~ m~_y m<=y k k k E ..,~ k 6 .:g/" m ~ y

= 2 Z 1>-_ 2(~-1)>-_ kE~/" n- -U n k E s r

<l ,n~- - k k

>_ ~(.. . ), >__ :.(..) =. ~, -- k k 8N/y k "-k-2-k T k -'k ~

2 ~']r 1 U y U y K t 4N/-----~ = 8---ff kcar2 1 = ~ > K

n 2N since for kEaC and n<=2N, -~<2---~=y.

6. In this section, we complete the proof of the theorem. By using (1), Lemmas 1, 5, 9, Cauchy's inequality and Parseval's formula, we obtain for large N that

(14)

/ F(~) G (~) U ( - ~) S ( - ~) = d~ + [ J - + / d~)[ ~IQF(~)G(a)S(_~) ( 1 U(-~) -~tQ U ,

1-1]Qf [ ~)d= +

+"~ I I <= f IF(=)IIG(cz)IlS(_=)I U-U(-cz ) d~+ -IlQ U

+ f IF(=)I[G(~)IIS(-~)I(I+ d~ <- f [F(~)IlG(~)IS d~+ +110. - I l Q

A c t a Ma themar H u n g a r i c a 44, 1084


+ 1-1 /Q +IIQ U Q

f If(~)lla(~)[( m a x IS(/ )l)2d <= f 1F(cz)lla(~)lgy4 d~+ + IIQ 1]Q<fl < I - 3-]Q - l l Q

1-3-1Q + f [F(a)]lG(a)ll4(gNl~ ~/22d~ <=

+IIQ

3_

Ky9N/y < 4 f K N ]1/2

I, log N)

}/ 1 4-28(KNlogN) ~/~ {( [F(a)]2d~)(f ]G(~)]2dR)} 112---

O

= 64 (KN log N) 1/2 (AB) ~/2 = 64 (ABKN log N) ~/2.

Furthermore, by (2), (3) and Lemma 10 and since h,>=O and %=>0, we have

1

(15) f F ( o : ) G ( a ) U ( - ~ ) S ( - a ) d o ~ : f f ~ hne(na)}12N+~U-3-v.e(-n~))da= 0 O ~,n=N+l \ n=3-

2N 2N 2N

Z h.v.>- Z h .K=g Z h.= n=N+3- n = N + l n = N + l

=K Z 1 : K ( Z 1)(Z 1)=ABK. N<a+b<=2N a E d b E ~

a E ,sur bEY3

(14) and (15) yield with respect to (5) that

1 1 1 ]JI >- -~ / F(a)G(a)U(-~)S(-c 0 da -64(ABKNlog N) 3-/2 >=

ABK ABK U 64(ABKNlog N) 1!2 > - - -

9N/y 64 (ABKN log N) a/2 =

[~ ABKye ]1/2 --576} 1 = 1 (ABKN log N) 1/2 [[N3 log N) > -ff (ABKNlog N) 1/2 (103 - 576) > 0

which proves (9) and this completes the proof of Theorem 1.

7. In this section, we derive the following corollary from Theorem 1.

COROLLARY 1. Let N>N3- and let d , g$ be sequences of integers such that

N/2 < a <- N (for aCd), N/2 < b <-_ N (for bE~) and (16) AB => 1 0 6 N a / 2 ( l o g N ) 2.

A c t a M a t h e m a t i c a Hungar ica 44, i984

176 A. BALOG AND A. SARKOZY

Then there exist integers a, b such that aC ~Y, bC N and

AB P(a + b) > 10 -6

N (log N) ~ "

Note that in the special case d =M, an old result of P. Erd6s and P. Turfin (see [2]) yields the much weaker estimate

max P(a + b) >> log IA]; a,b

on the other hand, this estimate holds without a condition of type (16).

N 2 (log N) ~ and let ~f'denote the set of prime numbers such PROOF. Put y=106 AB

that 2N/y<k<-4N/y. In order to show that Theorem 1 can be applied with this y, ~ , M and o~f, we have to show that (1), (5) and (6) hold.

Obviously, we have

N 2 (log N) 2 (17) Y = 106 AB

and in view of (16),

N ~ (log N) z (18) Y = 106 AB

(17) and (18) yield (1).

N 2 (log N) 2 _-> l0 s N~ - 106 (log N) 2 > 1

<_ 106 N 2 (log N) 2 = Na/2" - 106N a/2 (log N) ~

By the prime number theorem, for large N we have

(19) K = ~ 1 > _N/y N 2N[y.<p<=4NlY log N/y > -y log-----N"

Thus with respect to (16), we have

U U o@U ABKy ~ > A B l y = AB~g---g-~y = AB 106 N2(I~

which proves (5). Finally, in view of (19), kCJC implies that

p(k) = k > 2N/y -- 2N1/2K1/2 ( ) 1/z y (log N) 1/2 K >

- - IO"N a l o g N

N / ~/2 2N1/2 t y 1---]-o~g N) N.)1/2 =

Y ( l ~ 1/2 (.NlOgK

y3/2 log N (N1/~) ~/~ log N

log N

which proves that also (6) holds, so that in fact, Theorem 1 can be applied.

A c t a M a t h e m a t i c a Hungar i ca 44, 1984


By using Theorem 1, we obtain the existence of integers a, b, k such that a E d , bEN, kE~f and k](a+b). But this implies that

P(a + b) >= P(k) = k > 2N 2N

Y 10 G N2(log N) ~ AB

> 10 -8 AB N (log N) ~

which completes the proof of Corollary 1.

8. I f A>>N, B>>N, then Corollary 1 yields the existence of a, b such that a E d , bEN and

N (20) P(a + b) ~ (log N) 2 "

We conjecture that this assertion is true also with N on the right hand side; unfortu- nately, we have not been able to show this. However, by using a different method (based on the large sieve), we can show that the right hand side of (20) can be replaced

N by lo----g--N; furthermore, for small AB, the estimate obtained in this way is much better

than the one in Corollary 1.

THEOREM 2. I f N>N2, t i c { l , 2 . . . . ,N}, N o { l , 2 . . . . , N } and

(21) AB > 100 N(log N) 2

(where A = ~ 1, B = ~ 1) then there exist integers a, b such that aEag, bEN and a C d b E ~ 1 (AB) 1/2

P(a+b)> 16 l o g N "

PROOF. We use the following form of the large sieve:

LEMMA 11. Let X be a set of Z integers in the interval [ M + I , M + N ] . For prime p let co(p) denote the number of residue classes modulo p that contain no element of JU. Then for Q>=I

N + 2Q ~ Z < = - -

L where

#2(q) IT co (p) L = ~ ~<=a " ~l~P-co(P)

See [4], p. 25. In order to prove Theorem 2, put

and assume indirectly that (22)

1 (AB) l/2

Q =- 8 log N '

P(a + b) <- Q/2

12 A c t a M a t h e m a t i c a Hungar ica 44, 1984

1"/8 A. BALOG AND A. S~,.RKOZY

(for all aE~g and bE~) which implies that

(23) p~((a+b) for Q / 2 < p ~ Q , aE~, bE~.

For Q/2<p<=Q, let ~0(p) and ~(p) denote the number of residue classes modulo p that contain no element of d and ~ , respectively. If ~r contains an element of the residue class represented by the integer r, then by (23), & must not intersect the residue class represented by p - r. Thus we have ~ (p) =>p- ~p (p) hence

(24) ~0 (p) +. 0 (P) ~ P.

By using Lemma 11 with ~ ' and ~ , respectively, in place of J:, we obtain that

(25) A ~ N+2Q2 ~ N+2Q3 = N+2QZ

3 . . ~(p) r Z p3(q) I[ ~(p) - ~ P (P)p_--~p)

~<=Q plq P r Q]2<p~Q e/2<,<_-a p - 9 (P)

and in view of (24),

(26)

N + 2Q ~ B ~

(P) Z ~ ( q ) / / p - ~ ( p ) q~_Q t, lq

N + 2 Q 2 N + 2Q 3 3 . t~(p) - p--q)(p)"

Z P(P) p_-Z'~p) QI3<R<=Q Q/2.<p<=Q cp(p)

By using Cauchy's inequality and with respect to (21), we obtain from (25) and (26) that for large N (so that also Q is large by (21)),

(U + 2Q3) 2 AB ~

~o (p) V P - 'P (P) QI2-~p~_~Z p _ 9 (P) ~12--p~a ~ 9 (P)

(N+ 2Q3) 2

el2<'~p~e p_.'S-_-~p)) ( ~ - ~ J J

(N+ 2Q2)3 1 (N+ 2Q~)2 Q3 ( N ~ )~ -- < < 5Q 3 -~s (log0) n=

( Z 1) 3 5( logQ)~ QI3-<p~_Q

= 5 64 (log N ~ ) 3 AB ~-2 log 8 log N !

5 AB ( 6 4 N ( l o g N ) 2 / 3 < 64 (logN) 3 100N(logN) 3 +2 (logN) 3 < AB33 < AB,

and this contradiction shows that the indirect assumption (22) cannot hold which completes the proof of Theorem 2.

Aeta Mathematica Hungarica 44, 1984

ON SUMS OF SEQUENCES OF INTEGERS, II 179

References

[1] A. Balog and A. S~rkSzy, On sums of sequences of integers, I, Acta Arithmetica, to appear. [2] P. Erd6s and P. Tur~m, On a problem in the elementary theory of numbers, American Math.

Monthly, 41 (1934), 608--611. [3] L. K. Hua, Additive Primzahltheorie, Teubner (Leipzig, 1959). [4] H. L. Montgomery, Topics in MMtiplicative Number Theory. Springer-Verlag (1971).

(Received October 26, 1982)

MATHEMATICAL INSTITUTE OF THE H U N G A R I A N ACADEMY OF SIENCES BUDAPEST, R E ~ L T A N O D A U. 13--15 H- - I053 H U N G A R Y

12" A c t a M a t h e m a t i c a H u n g a r i c a 44, 1984

on sums of sequences of integers. ii

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