The Maximum Upp er Density of a Set of Positive

Real Numb ers with no solutions to x + y = kz

John L. Goldwasser

West Virginia University

Morgantown, WV 26506

Fan R. K. Chung

UniversityofPennsylvania

Philadelphia, PA 19104

January 11, 1996

Abstract

If k is a p ositive real numb er, wesay that a set S of real numb ers is k -sum-free if there do

not exist x; y ; z in S such that x + y = kz .For k  4we nd the maximum upp er densityof

a k-sum-free subset of the set of p ositive real numb ers. We also show that if k is an integer

greater than 3 then the set of p ositive real numb ers and the set of p ositiveintegers are each

the union of three but not two k -sum-free sets.

1. Intro duction

Wesay that a set S of real numb ers is sum-free if there do not exist x; y ; z in S such that

x + y = z .Ifkis a p ositive real numb er, wesay that a set S is k -sum-free if there do not

exist x; y ; z in S such that x + y = kz we assume not all x, y , and z are equal to each other

to avoid a triviality when k = 2. Many problem in group theory and numb er theory fo cus on

sum-free sets. In work related to Fermat's Last Theorem, Schur [Sc] proved that the p ositive

integers cannot b e partitioned into nitely many sum-free sets. Van der Warden [W] proved

that the p ositiveintegers cannot b e partitioned into nitely many 2-sum-free sets.

If S is a subset of the p ositiveintegers we de ne the upp er density  S  and lower density

U

n o

jS \f1;2;:::;ngj

+

 S ofS to b e the limit sup erior and limit inferior resp ectively of j n 2 Z .If

L

n

kis a p ositiveinteger let U k  and Lk  denote the supremum of  S  and  S  resp ectively

U L

over all k -sum-free subsets S of the p ositiveintegers. Let f n; k  b e the maximum size of

a k -sum-free subset of f1; 2;:::;ng and let Gk  denote the limit sup erior over the p ositive

f n;k 

. For any k we clearly have the relationship Lk   U k   Gk . Since integers of

n

1 1

the o dd p ositiveintegers are sum-free, L1  . It is easy to show that G1  ,so

2 2

1

L1 = U 1 = G1 = . Roth [Ro] showed that G2 = 0. His results were strengthened by

2

Szemer edi [Sz], Salem and Sp encer [SS], and Heath-Brown [H].

If k is a p ositiveinteger and S is a k -sum-free subset of the p ositiveintegers with x 2 S

1

and y 2 S \f1;2;:::;kxg, then kx y 62 S ,soLk .Ifkis o dd then, since the o dd

2

1

integers are then k -sum-free, Lk = .Ifk6=2iseven, then the set of all p ositiveintegers

2

n o

k2 1 1 1

whose mo d k congruence class is in 1; 2;:::; is k -sum-free. Hence  Lk   for

2 2 k 2

even k .

Chung and Goldwasser [CG1] showed that if n  23 then the set of all o dd p ositive

integers less than or equal to n is the unique maximum 3-sum-free subset of f1; 2;:::;ng.

1

Hence L3 = U 3 = G3 = .

2

The ab ove density functions have analogs over k -sum-free subsets of the p ositive real num-

b ers where k is any p ositive real numb er. If S is a Leb esgue measurable k -sum-free subset of

the p ositive real numb ers, we de ne the upp er density  S  and lower density  S ofS to b e

u l

n o

S \0;x]

+

the limit sup erior and limit inferior resp ectively of j x 2 R where  denotes mea-

x

sure. Let uk  and l k  denote the least upp er b ound of  S  and  S  resp ectively over all

u l

measurable k -sum-free subsets S of the p ositive real numb ers. Let g k  denote the maximum

size of a measurable k -sum-free subset of 0; 1]. Clearly wehave lkukgk for any

1

p ositive real number k. Itisobvious that g 1 = and can b e shown that g 2 = 0. Chung

2

and Goldwasser [CG2] found g k  for all k  4 and showed that there is an essentially unique

maximum set, the union of three intervals:

e ;f ] [ e ;f ] [ e ;f ]

1 1 2 2 3 3

where

2

4 2k 2

f = f = f =1 1.1

1 2 3

4 2 4 2

k 2k 4 k 2k 4

2

and e = f i =1;2;3.

i i

k

In this pap er we will nd uk  for k  4. We will generalize a result of Rado [R , R1]

by showing that for any p ositiveinteger k greater than 3 the p ositive real numb ers and the

p ositiveintegers are each the union of three but not two k -sum-free sets and that the p ositive

real numb ers and the p ositiveintegers are each the union of four but not three 3-sum-free

sets. 2

2. Maximum upp er densityofa k-sum-free set

2

Lemma 1. Suppose k  4 is a real number, c and w arepositive real numbers with c  w ,

2

k

and S isameasurable k -sum-free subset of the positive real numbers which contains c. Then

c 2 2

w + ;w  1 w;  S \

2

k k k

with equality if and only if

2 2

 S \ w; w = 1 w:

k k

2 1

Pro of. Let S b e a set satisfying the hyp otheses and supp ose S \ w; w 6= . If x 2

k k

2 1

S \ w; w then 0

k k

[kx w; w]: if z 2 S \ [kx w; w] then kx z 2 [kx w; w] but kx z 62 S . Hence

1

S \ [kx w; w]  [w kx w] 2.1

2

n o

x 1 2

and if y is the infemum of j x 2 S \ w; w then

w k k

1

S \ [ky w w; w]  [w ky w w] : 2.2

2

1 2 2 c 2 c

If there exists x 2 S \ w; w such that kxw  w+ then, letting v =  S \ w + ;w ,

2 2

k k k k

k k

by 2.1

1 1 2

v  [w kx w] < w  1 w;

2 2 k

2 c 2 c

so the conclusion of the Lemma holds if ky w w< w+ .Ifky w w  w + , there are

2 2

k k

k k

three cases to consider.

1 1 1

Case i. Supp ose  y  . Then ky w w  yw and, since S \ w; y w = ,

k k 1 k

 

2 c 1

v =  \ w + ; min w; ky w w + S \ ky w w; w

2

k k k

 

1 2 c 1

 min w; ky w w w + + [w ky w w] : 2.3

2

k k k 2

1

The right-hand side of 2.3 is clearly a maximum when ky w w = w,so

k

 

2

1 2 c 1 1 k +k4 c

v  w w+ + w w = w

2 2

k k k 2 k 2k k

2

c k +k4+k 4k 1 2 c

w  = 1 w

2

2k k k k

2

< 1 w:

k 3

1 1 2

Case ii. Supp ose

2

k 1 k

k

c 1 2

w + v =  S \ ; w + S \ yw; kyw w + S \ ky w w; w

2

k k k

c 1 1 2

w w + +ky w w yw + [w ky w w] 

2

k k k 2

2 c 2 2 c k 2 1 k

1 y + w  + + w =

2 2 2

2 k k 2 k k k k

k 2 c k 2 c 2 k +4

= w  w < 1 w:

k 2k k k k k

1 2 2

Case iii. Supp ose y> + and v = a + b + d + e where a = . Then ky 1 >

2

k k

k

2 1 2 c 2

 S \ ; w , b =  S \ yw; w , d =  S \ w; w c , and e = S \ w w +

2

k k k k

k

2

c; w . Then b  y w and by 2.2.

k

2 1

d + e  1 w [w ky w w]

k 2

2 k 2 2 k

= 1 w y w  1 w b:

k 2 k k 2

Hence

k 2

b + +e  1 w: 2.4

d

2 k

2 c 1 2

Since c 2 S ,if x2S\ w+ ; w then w

2

k k k

k

forbidden pairing,

2

ka + d  w c w: 2.5

k

Thus wehave

d k d e e k

a+ + b+ + + v 

2 2 4 2 2 2

 

1 2 2 1 2 c

 = 1 1 w c + 1 w + w 2.6

2 k 2 k 2 k

by 2.4, 2.5, and the fact that e  c.For equality to hold in the Lemma, wemust b e in

Case iii and 2.4, 2.5, and 2.6 all must b e equalities, which completes the pro of.

Theorem 1. If k  4 is a positive real number and S is a measurable k -sum-free subset of the

2

k 2k

. positive real numbers, then the upper density of S is at most

2

k 2

Pro of. Let S b e a measurable k -sum-free subset of the p ositive real numb ers containing c

m

2 2

and z where c  z . Let m b e the largest p ositiveinteger such that z  c, that is

2 2

k k

let

 $

c

log

z

; m =

2

log

2

k 4

i

2

z for i =0;1;:::;m. By Lemma 1, and let w =

i

2

k

2 c 2

 1  S \ w ;w w + for i =0;1;:::;m 1 :

i i i

2

k k k

Hence

! 

m1

[

2

w ;w S \ 0;z] = S \ 0;w  +  S \

i i m

2

k

i=0

m1

i

X

2 c 2

 w + z 1 + m

m

2

k k k

i=0

and

 

m

2 2

k m S \ 0;z] c 2 k 2k c

 + 1 + :

2 2

z 2 z k k 2 k z

Taking the limit as z go es to in nity so m go es to in nity since c is xed then gives the

result.

The set

 

i i

[

2 2 2

T 1= ; 2.7

k

2 2

k k k

i2Z

2

k 2k

has upp er density for k>2 so the b ound in Theorem 1 is b est p ossible. Hence for any

2

k 2

2

8k 2

k 2k

real number k greater than 2, g k   + the three intervals de ned in

2 2 4 2

k 2 k k 2k 2k 4

2

k 2k

, and these are b oth equalities if k  4. equations 1.1, uk  

2

k 2

These constructions can b e used to pro duce k -sum-free subsets of the p ositiveintegers as

well. Let k b e a p ositiveinteger greater than 2 and let J b e the union of the three intervals

k

de ned in equations 1.1. De ne a subset H nof f1;2;:::;ng by H n=f1;2;:::;ng\

k k

fnx j x 2 J g and a subset H 1 of the p ositiveintegers Z by H 1= Z\T 1. Then

k k k k

2 2

jH nj

k 2k k 2k

k

lim = J  and  H 1 = ,soGk J and U k   . We

k U k 2 k 2

n

k 2 k 2

n!1

conjecture that these are b oth equalities if k is an integer greater than 3 not for k = 3 b ecause

1 1

these values are then b oth less than and the o dd integers givevalues equal to . Theorem 1

2 2

2

k 2k

do es not apply for 2

2

k 2

and Goldwasser discuss some conjectures ab out g k  for 2

3. The p ositiveintegers and real numb ers as the union of k -sum-free sets

In contrast to the situation for k = 1 and k =2,if k is an integer greater than 2 then the

p ositive real numb ers and the p ositiveintegers are each the union of nitely many k -sum-free

sets. 5

Theorem 2. If k is an integer greater than or equal to 4 then the positive integers and the

positive real numbers areeach the union of three k -sum-free sets, but not of two. The positive

integers and the positive real numbers areeach the union of four 3-sum-free sets, but not of

three.

Pro of. First we show that for any p ositiveinteger k , the set of all p ositiveintegers, and

hence the set of all p ositive real numb ers, is not the union of two k -sum-free sets. Supp ose A

and B are k -sum-free sets whose union is the p ositiveintegers. Let x b e an integer such that

kx 2 A and k x +1 2 B suchaninteger exists b ecause the multiples of k greater than any

xed numb er is not a k -sum-free set and let y b e an integer greater than x such that y 2 A

and y +1 2 B. Then k y x= ky kx = ky +1kx+ 1 cannot b e in either A or B .

If x is any p ositive real numb er and k  3we de ne the k -sum-free set T x; 1by

k

T x; 1= fxy j y 2 T 1g :

k k

2

k

where T 1 is de ned in 2.7. We note that T ; 1 = T 1. If k  4 then T 1 [

k k k k

2

2 2

k k k

T ; 1 [ T ; 1 is the set of all p ositive real numb ers b ecause  k if k = 4 the three

k k

2 4 4

sets are actually disjoint. Thus for k  4 the p ositive reals, and hence the p ositiveintegers,

are the union of three k -sum-free sets. If k = 3 the ab ove union of three sets do es not cover

2 3

3 k

the p ositive real numb ers b ecause < 3. However, if we add T ; 1 to the union, then

k

4 8

3

3

the four sets do cover the p ositive reals b ecause > 3. To complete the pro of we need only

8

show that the set of p ositiveintegers is not the union of three 3-sum-free sets. This could b e

done by a direct argument with many cases. Instead wechose to do it by computer. It turns

out that 53 is the largest integer n such that hte set of p ositiveintegers less than or equal to

n is the union of three 3-sum-free sets. One such partition is

A = f1; 3; 4; 7; 10; 12; 13; 16; 19; 21; 22; 25; 28; 30; 31; 34; 37; 39; 40; 43; 46; 48; 49; 52g;

B = f2; 5; 6; 15; 18; 24; 33; 35; 38; 41; 42; 44; 45; 47; 50; 51; 53g;

C = f8; 9; 11; 14; 17; 20; 23; 26; 27; 29; 32; 36g :

4. Op en problems

1

S

[3i 2; 3i 1 is a sum-free subset of the p ositive reals with upp er and lower The set

i=1

1 1

density equal to . Hence  l 1  u1.

3 3 6

1

Conjecture 1. l 1 = u1 = .

3

We restate here the conjectures of Section 2:

Conjecture 2. Gk = gk and U k = uk for k  4.

2

k 2k

Conjecture 3. uk = for k>2.

2

k 2

1 1

The b ound  Lk  for the maximum lower densityofak-sum-free subset of the

2 k

p ositive itnegers for even k 6= 2 is not b est p ossible. One can improve it slightly by considering

t

integers  mo d k  for various p ositiveintegral values of t, but we do not have a conjecture as

to the actual value of Lk  for even integers greater than 4. The p ositiveintegers congruent

2

to 2 or 3  mo d 5 or congruent to 1 or 4  mo d 5 are 4-sum-free, so L4  .

5

2

Conjecture 4. L4 = .

5

1 k 2

for k  3, whichis times its upp er density. The set T 1 has lower density

2 k

k

k 2

One can obtain a k -sum-free set S of the real numb ers for k  3by uniformly \fattening

up" the o dd integral p oints as much as p ossible and translating the resulting set. The set

1

S

2 2 1

+k +2i; +k+2i+1 has lower and upp er density if k  3, which

k 2 k 2 k +2

i=0

k 2

is a little more than .

2

k 2

1

Conjecture 5. l k = for k 6=2.

k+2

Conjecture 6. If k<4 the positive real numbers are not the union of three k -sum-free sets.

Of course Theorem 2 shows Conjecture 6 is true for k  3.

References

[CG1] F. R. K. Chung and J. L. Goldwasser, Integer sets containing no solutions to x + y =3z,

The Mathematics of Paul Erd}os, R. L. Graham and J. Nesetril eds., Springer-Verlag,

Heidelb erg, 1996.

[CG2] F. R. K. Chung and J. L. Goldwasser, Maximum subsets of 0; 1] with no solutions to

x + y = kz , Electronic Journal of Combinatorics 3 1996.

[H] D. R. Heath-Brown, Integer sets containing no arithmetic progressions, J. London

Math. Soc. 2 35 1987, 385{394. 7

[R] R. Rado, Verallgemeinerung eines Satzes von van der Waerden mit Anwendungen auf

ein Problem der Zahlentheorie, Sitzungsber. Preuss. Akad. Berlin 27 1933, 3{10.

[R1] R. Rado, Studien zur Kombinatorik, Math. Zeit. 36 1933, 424{480.

[Ro] K. Roth, On certain sets of integers, J. London Math. Society, 28 1953, 104{109.

m m m

[Sc] I. Schur, Ub er die Kongruenz x + y = z mod p, J. ber. Deutch. Math.-Verein

25 1916, 114{116.

[SS] R. Salem and D. C. Sp encer, On sets of integers which contain no three terms in

arithmetical progressions, Proc. Nat. Acad. Sci. USA 28 1942, 561{563.

[Sz] E. Szemer edi, On sets of integers containing no k elements in arithmetic progression,

Acta Arith. 27 1975, 199{245.

[W] B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk 15

1927, 212{216. 8