The Maximum Upper Density of a Set of Positive Real Numbers with no solutions to x + y = z John L. Goldwasser West Virginia University Morgantown, WV 6506 Fan R. K. Chung University ofpennsylvania Philadelphia, PA 904 January, 996 Abstract If is a positive real number, we say that a set S of real numbers is -sum-free if there do not exist x; y; z in S such that x + y = z. For 4we nd the maximum upper density of a-sum-free subset of the set of positive real numbers. We also show that if is an integer greater than 3 then the set of positive real numbers and the set of positive integers are each the union of three (but not two) -sum-free sets.. Introduction We say that a set S of real numbers is sum-free if there do not exist x; y; z in S such that x + y = z. If is a positive real number, we say that a set S is -sum-free if there do not exist x; y; z in S such that x + y = z (we assume not all x, y, and z are equal to each other to avoid a triviality when = ). Many problem in group theory and number theory focus on sum-free sets. In wor related to Fermat's Last Theorem, Schur [Sc] proved that the positive integers cannot be partitioned into nitely many sum-free sets. Van der Warden [W] proved that the positive integers cannot be partitioned into nitely many -sum-free sets. If S is a subset of the positive integers we dene the upper density U (S) and lower density L (S) ofsto be the limit superior and limit inferior respectively of n js\f;;:::;ngj n j n Z + o.if is a positive integer let U() and L() denote the supremum of U (S) and L (S) respectively over all -sum-free subsets S of the positive integers. Let f(n; ) be the maximum size of a -sum-free subset of f; ;:::;ng and let G() denote the limit superior over the positive integers of f (n;). For any we clearly have the relationship L() U() G(). Since n the odd positive integers are sum-free, L(). It is easy to show that G(),so
L() = U() = G() =. Roth [Ro] showed that G() = 0. His results were strengthened by Szemeredi [Sz], Salem and Spencer [SS], and Heath-Brown [H]. If is a positive integer and S is a -sum-free subset of the positive integers with x S and y f;;:::;xg, then x, y 6 S, sol(). If is odd then, since the odd integers are then -sum-free, L() = n. If 6=iseven, then the set of all positive integers o whose mod congruence class is in ; ;:::;, is -sum-free. Hence, L() for even. Chung and Goldwasser [CG] showed that if n 3 then the set of all odd positive integers less than or equal to n is the unique maximum 3-sum-free subset of f; ;:::;ng. Hence L(3) = U(3) = G(3) =. The above density functions have analogs over -sum-free subsets of the positive real numbers where is any positive real number. If S is a (Lebesgue) measurable -sum-free subset of the positive real numbers, we dene the upper density u (S) and lower density l (S) ofsto be the limit superior and limit inferior respectively of n (S\(0;x]) x j x R + o where denotes measure. Let u() and l() denote the least upper bound of u (S) and l (S) respectively over all measurable -sum-free subsets S of the positive real numbers. Let g() denote the maximum size of a measurable -sum-free subset of (0; ]. Clearly we have l() u() g() for any positive real number. Itisobvious that g() = and can be shown that g() = 0. Chung and Goldwasser [CG] found g() for all 4 and showed that there is an essentially unique maximum set, the union of three intervals: (e ;f ][(e ;f ][(e 3 ;f 3 ] where f = and e i = f i (i =;;3). 4 4,, 4 f = (, ) 4,, 4 f 3 = (.) In this paper we will nd u() for 4. We will generalize a result of Rado [R, R] by showing that for any positive integer greater than 3 the positive real numbers and the positive integers are each the union of three (but not two) -sum-free sets and that the positive real numbers and the positive integers are each the union of four (but not three) 3-sum-free sets.
. Maximum upper density of a -sum-free set Lemma. Suppose 4 is a real number, c and w are positive real numbers with c w, and S isameasurable -sum-free subset of the positive real numbers which contains c. Then w + c ;w, w; with equality if and only if w; w =, w: w; w 6=. If x w; w then 0 <x,w<wand there is a \forbidden pairing" with respect to x of Proof. Let S be a set satisfying the hypotheses and suppose [x, w; w]: if z [x, w; w] then x, z [x, w; w] but x, z 6 S. Hence and if y is the infemum of If there exists x S\ by (.) ( [x, w; w]) [w, (x, w)] (.) o w j x w; w then n x ( [yw, w; w]) [w, (yw, w)] : (.) w; w such that x,w w+ c then, letting v = S \ v [w, (x, w)] < w, w; w + c ;w, so the conclusion of the Lemma holds if yw, w< w+ c.ifyw, w w + c, there are three cases to consider. Case (i). Suppose y. Then yw, w yw and, since, w; yw =, v = \ w + c ; min w; yw, w + ( (yw, w; w)) min w; yw, w, w + c + [w, (yw, w)] : (.3) The right-hand side of (.3) is clearly a maximum when yw, w = w,so v w, w+c + w, w = +,4 w, c +,4+(,4)(, ) w, c =, w, c <, w: 3
Case (ii). Suppose <y, +. Then y<y, and v = w + c + ( (yw; yw, w)) + ( (yw, w; w)) = w, w + c, y+ =, ; w +(yw, w), yw + [w, (yw, w)] w, c, +4 w, c, w, c <, Case (iii). Suppose y > +. Then y, > c; w)). Then b Hence w + c ; w, b =, y w and by (.). d + e Since c S, ifxs\ forbidden pairing, =,, + + yw; w, d = w, [w, (yw, w)] w,, y w w: w, c and v = a + b + d + e where a = w; w, c, and e = ( (w,, w, b: b + d +e, w: (.4) w+ c ; w then w<x,c<w,cbut x, c 6 S. By this a + d w, c, w: (.5) Thus we have v a+d, + 4 b+d +e w, c + + e, w + c =, w (.6) by (.4), (.5), and the fact that e c. For equality to hold in the Lemma, we must be in Case (iii) and (.4), (.5), and (.6) all must be equalities, which completes the proof. Theorem. If 4 is a positive real number and S is a measurable -sum-free subset of the positive real numbers, then the upper density of S is at most,,. Proof. and z where c let Let S be a measurable -sum-free subset of the positive real numbers containing c z. Let m be the largest positive integer such that m z c, that is m = $ % log c z log 4 ;
i and let w i = z for i =0;;:::;m. By Lemma, w i;w i, w i + c for i =0;;:::;m, : Hence ( (0;z]) = ( (0;w m )) + w m +z, m, X i=0 m, [ i=0 i + m c w i;w i! and ( (0;z]) z c m z +,,, + m Taing the limit as z goes to innity (so m goes to innity since c is xed) then gives the result. The set has upper density,, T () = [ iz i ; i # c z : (.7) for > so the bound in Theorem is best possible. Hence for any real number greater than, g(),, + 8(,) (,)( 4,,4) equations (.)), u(),, and these are both equalities if 4., (the three intervals dened in These constructions can be used to produce -sum-free subsets of the positive integers as well. Let be a positive integer greater than and let J be the union of the three intervals dened in equations (.). Dene a subset H (n) off;;:::;ng by H (n)=f;;:::;ng\ fnx j x J g and a subset H () of the positive integers Z by H () =Z\T (). Then jh lim (n)j = (J n! n ) and U (H ()) =,,so G() (J, ) and U(),. We, conjecture that these are both equalities if is an integer greater than 3 (not for = 3 because these values are then both less than and the odd integers give values equal to ). Theorem does not apply for <<4, but we suspect U() =,, for these values of as well. Chung and Goldwasser discuss some conjectures about g() for <<4 in [CG]. 3. The positive integers and real numbers as the union of -sum-free sets In contrast to the situation for = and =,ifis an integer greater than then the positive real numbers and the positive integers are each the union of nitely many -sum-free sets. 5
Theorem. If is an integer greater than or equal to 4 then the positive integers and the positive real numbers are each the union of three -sum-free sets, but not of two. The positive integers and the positive real numbers are each the union of four 3-sum-free sets, but not of three. Proof. First we show that for any positive integer, the set of all positive integers, and hence the set of all positive real numbers, is not the union of two -sum-free sets. Suppose A and B are -sum-free sets whose union is the positive integers. Let x be an integer such that x A and (x + )B (such aninteger exists because the multiples of greater than any xed number is not a -sum-free set) and let y be an integer greater than x such that y A and y +B. Then (y, x) =y, x = (y +),(x+ ) cannot be in either A or B. If x is any positive real number and 3we dene the -sum-free set T (x; ) by T (x; ) =fxy j y T ()g : where T () is dened in (.7). We note that T T ; [T ; = T (). If 4 then T () [ ; 4 is the set of all positive real numbers because (if = 4 the three 4 sets are actually disjoint). Thus for 4 the positive reals, and hence the positive integers, are the union of three -sum-free sets. If = 3 the above union of three sets does not cover the positive real numbers (because 3 4 < 3). However, if we add T the four sets do cover the positive reals (because 33 8 3 8 ; to the union, then > 3). To complete the proof we need only show that the set of positive integers is not the union of three 3-sum-free sets. This could be done by a direct argument with many cases. Instead we chose to do it by computer. It turns out that 53 is the largest integer n such that hte set of positive integers less than or equal to n is the union of three 3-sum-free sets. One such partition is A = f; 3; 4; 7; 0; ; 3; 6; 9; ; ; 5; 8; 30; 3; 34; 37; 39; 40; 43; 46; 48; 49; 5g; B = f; 5; 6; 5; 8; 4; 33; 35; 38; 4; 4; 44; 45; 47; 50; 5; 53g; C = f8; 9; ; 4; 7; 0; 3; 6; 7; 9; 3; 36g : 4. Open problems The set S [3i, ; 3i, ) is a sum-free subset of the positive reals with upper and lower i= density equal to. Hence l() u(). 3 3 6
Conjecture. l() = u() = 3. We restate here the conjectures of Section : Conjecture. G() =g()and U() =u()for 4. Conjecture 3. u() =,, The bound, for >. L() for the maximum lower density ofa-sum-free subset of the positive itnegers for even 6= is not best possible. One can improve it slightly by considering integers (mod t ) for various positive integral values of t, but we do not have a conjecture as to the actual value of L() for even integers greater than 4. The positive integers congruent to or 3 (mod 5) (or congruent to or 4 (mod 5)) are 4-sum-free, so L(4) 5. Conjecture 4. L(4) = 5. The set T () has lower density,, for 3, which is times its upper density. One can obtain a -sum-free set S of the real numbers for 3by uniformly \fattening up" the odd integral points as much as possible and translating the resulting set. The set S i=0, +(+)i;, +(+)i+ has lower (and upper) density + is a little more than,,. Conjecture 5. l()= + for 6=. (if 3), which Conjecture 6. If <4the positive real numbers are not the union of three -sum-free sets. Of course Theorem shows Conjecture 6 is true for 3. References [CG] 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, Heidelberg, 996. [CG] [H] F. R. K. Chung and J. L. Goldwasser, Maximum subsets of (0; ] with no solutions to x + y = z, Electronic Journal of Combinatorics 3 (996). D. R. Heath-Brown, Integer sets containing no arithmetic progressions, J. London Math. Soc. () 35 (987), 385{394. 7
[R] R. Rado, Verallgemeinerung eines Satzes von van der Waerden mit Anwendungen auf ein Problem der Zahlentheorie, Sitzungsber. Preuss. Aad. Berlin 7 (933), 3{0. [R] R. Rado, Studien zur Kombinatori, Math. Zeit. 36 (933), 44{480. [Ro] K. Roth, On certain sets of integers, J. London Math. Society, 8 (953), 04{09. [Sc] [SS] [Sz] I. Schur, Uber die Kongruenz x m + y m = z m (mod p), J. ber. Deutch. Math.-Verein 5 (96), 4{6. R. Salem and D. C. Spencer, On sets of integers which contain no three terms in arithmetical progressions, Proc. Nat. Acad. Sci. USA 8 (94), 56{563. E. Szemeredi, On sets of integers containing no elements in arithmetic progression, Acta Arith. 7 (975), 99{45. [W] B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wis 5 (97), {6. 8