Dressler, Robert E.

Pigno, Louis

English

Dressler Robert E.

Pigno Louis

University of Szeged

Acta Sei. Math., 47 (1984), 233—200 Small sum sets and the Faber gap condition ROBERT E. DRESSLER and LOUIS PIGNO*) . Our purpose in, this paper is to analyze sets satisfying a small sum set condition in terms of the classical Pa te r gap condition and to prove a "shape" result for sets of positive upper Banach density., For simplicity's sake we will take all sequences (before Theorem 4) to be increasing sequences of non-negative integers. As usual, Z + stands for the set of positive integers. : 1 ; D e f i n i t i o n 1 (The Faber gap condition of orderp). If 5 , = {J7}J11 and p£Z+, then S is said to be an F,^ set if s^^.p—sn— 1 D e f i n i t i o n 2 (Lacunarity condition L„). A set S is said to satisfy condition Lx if for every infinite sequence {«J}~=1, |lim (S—nj)\ is finite. Proceeding inductively, we say that S satisfies condition L„ if for every sequence {«j}~=1 the set lim (S—n)) satisfies condition /,„_{. D e f i n i t i o n 3 (Generalized Faber gap condition F(p"y). If S is an L„ set that is not Ln_x and for some fixed p every n fold iterated lim inf has no more t h a n e l e -ments, we say S has property F^K D e f i n i t i o n 4- If 2, then a set 5 is said to be a f?„ set if sup {min (l^xl, \A2], ...,\J„\): A + Examples of Q„ sets are constructed in [4] and [5]; see also [3]. Notice that any G„ set («¿2 ) is an i£,„_x set. Gur first theorem shbws that conditions Fp and Fjp coincide for all p. T h e o r e m 1. S J> an Fp set if and only if S is an F^ set. P r o o f , Assume only for the sake of notation that P S 2 . We will show that S~ is an Fp set but not an Fp_x set if and only if sup llim (S—JJ")1=p, where the sup is taken over all subsequences cr Received October.6, 1982 and in revised form March 10, 1983. *) The second author was supported by NSF grant MCS-8102061. 234 R . E . Dress ie r and L. P i g n o First, suppose S is an Fp set, but not an Fp_x set. Then by the pigeon hole prin-ciple there is a subsequence i such that for each t with 1 ^t^p—1, sJ+t—sji is constant for all /. It follows that {0,sh+1-sh,sh+2-sH, + i <= S which implies that [ l i m O n the other hand, if | l i m ( S — s ^ \ > p for some {sj)cS, then there exist A1<Xo<...<Xp + 1 such that, for some iV£Z+ and each t with x,+Sj£S for all i^N. Clearly then S is not an Fp set. Now suppose sup |lim (S—jy )| —p. If S were not an Fp set, then there would be N £ Z + and a subsequence j d S such that Sj_+p—Sj<N for all /'. An argu-ment similar to one we used above shows that |lim (S—jy )| =/>+1 which is a contra-diction. In addition, if and ^ < ^ < . . . < 1 , are such that for each t with l^t^p, -SjdS for all sufficiently large /, then S is not an Fp_1 set. The next theorem relates the Q2 property and the Fp property. T h e o r e m 2. If S is a Q2 set with sup {min fl^l, \A2\): A1 + As(^S}=p, then S is an Fp set. Proof . If S were not an Fp set, then there would exist N£ Z + and a subsequence {iy J ^ j C S such that Sj +p—Sj <N- for all i. As in the proof of Theorem 1, there are infinitely many i's such that for each / with l^tSp, sJi+t—Sj is constant for all such i's. Thus, {0, Sj.+x-Sj^Sj^i-Sj,, ...,Sjl+p-Sj} + {sj}lL1 c S and so sup .{min fl^l, \A2\): A1+A2aS}^p+1 which is a contradiction. Next we relate the g„ (n ̂ 2 ) property and the iterative property F£B). T h e o r e m 3. Let n^3. If S is a Q„ set with sup {min I^sl I^BI): and S is not an L„_2 set then S is an F(p~1} set. Proo f . Suppose S is as in the hypothesis and that, in contradiction of the con-clusion, the cardinality of some n — l fold iterated l iminfof S (with respect to se-quences {^5fc))n=i' 0=k=n—2) is greater than p. Then there are y^y**^..-i€Z+ such that for each t with 1h§/=§/>+1, v,+^"~2 )glimX X (• • .(Hm (iim ( S - s f ] ) • • -s(jn~0))) for all sufficiently large i: Next, for each t with + j ' , + ^ J 2 ) + 4 , ; - s ) e l i m (... ( l i m ^ - ^ ) - ^ ) - . . . - ^ - 4 » ) for all sufficiently large /, etc. Finally, we have, for each t with l^t^p+l, yt+stf-z)+s(?-3)+...+sf> + i-sf^S for all sufficiently large /. This contradicts the hypothesis that S is q„ and we are done. Smal l s u m sets a n d the F a b e r g a p condi t ion 235 Note. Theorem 3 tells us that if we have a g„ set that is not Ln_2 then whenever we take n—2 lim infs we have arrived at either an Fp set or a finite set. The next theorem, of interest in itself, and whose proof may remind the reader of Lukomskaya's proof of van der Waerden's Theorem in [6], will help us relate density to the F ^ property. We need the following definitions. De f in i t i on 5. If EczZ, then the upper Banach density of E is defined by S5(£)=l imsup( |Ff l / | ) / | / | , where / ranges over all bounded intervals in Z. i/l— Theorem 4. Let If Bd(E)>-0 then for each j~0, there exists etj, Mj and kj such that K- -,eij+Mj} + kj c {eij+l, ..., e;,+1+M.tl} with 0<Mj<MJ+i, and, for jx^j2, {elhi ..., eih+Uj)D{e(Jt, ..., ei)t+Uj) = 0. " In addition, there exists M > 0 such that for each j, Mj + 1 M " Proof . Say Bd(E)>l/N0 for some Ar06Z+. Then there exist infinitely many integers XQ such that |£T)[x0, xa+Na]\^NJ2Na=\l2. Form at most 2N«+1-\ classes of such intervals [x0, x0+iV0] according to the "shape" of £Tl[x0, x0+iV0]. Call such a class by the generic name C. At least one C„ is infinite. Next, choose iVi£Z+ such that iV0 divides Nx and NJ2N0>N0+1. Now there exist infinitely many integers Xx such that ¿¿Tlta, xl+N1]^NJ2N0. We take such intervals that contain a member of some class C0. Notice that by breaking up inter-vals of length Nx into consecutive intervals of length NQ, we see that, except for finitely many intervals, each interval of length Nx which has at least NJ2N0 members of E must contain at least one member of one class C„ (for if not, all but finitely many such intervals of length Nx would have fewer than (NJN0)(N0/2N0)=NX/2N0 mem-bers). Since there exist infinitely many intervals of length Nx which contain a member of some Cx we must have infinitely many such intervals containing infinitely many members of the same class C0. In addition, if we classify these intervals [Xx, x1+Ar1] according to the "shape" of F T l ^ j , ^-(-A^J we see that at least one such class of intervals is infinite. Call such classes of intervals of length Nx by the generic name Cx- Notice that since Ar1/2Ar0>iV0+1, we see that the intersection of E with any member of any Cx has more elements than the intersection of E with any member of any C„. Now choose N2£Z+ such that Nx divides N2 and N2/2N0>N1-H. We repeat the construction to obtain at least one infinite class C2 of intervals of the form 236 R . E . D r e s s i e r a n d L . P i g n o [x^Xz+N^ all of whose members have the same "shape" when intersected with E and, for some class Cx, each of whose members contains a member of C\ . Continue in this manner. Notice that at the f h stage, starting with y=0, only finitely many of theintervals [xj, XJ+NJ] which contain at least a proportion 1/2Nj of elements of E do not belong to some C,. Also, no two classes C, need be disjoint. , We thus may obtain, for each y'SO, an interval Xj+Nj]£Cj such that the intervals are pairwise disjoint and such that for each j ^ O there is a Ar7£Z with Xj + N j ] ) + k j C ( £ N [ X , + 1 , . Y J + 1 + JV , . + 1 ] ) • where since Nj+1/2N0>Nj +1, \EC][xj, Xj + JV,-]| < \EH[xj+1, xJ+1 + Nj+l]|. If we write {e, , ...,' ei +Mj]—E fl[x7-, XJ+Nj], the proof is complete. C o r o l l a r y . If E is an L„ set for some n then Bd(E) — 0. Proo f . If Bd(E)>0, then if e0 is the smallest element of _En[.r0> v0+iV0] and {" j j j lo^^o^ eo+k0, e0->rk0 + kx, ...} then hm(.E—nj) contains the set Bi = {0, e2-et, e3-e1, ..., ero-e1; e'rJrX-e\, ... p' _ p' p(2) _p<2) i>(2)_„(2) ) where EC1 [x0, *o + No) = {<?i, ..., e j , En[xx,xx + Nx) = {ei, ...,<}, Ef)[xj, XJ + NJ] = {EP, By construction Bd(Bx)>0 because for each j, ..., e^} contains at least a proportion IJ2N0 of the interval [xj, Xj+Nj]. Now, perform the construction another n — 1 times to obtain a set Bn with Bd(B„)=~0. Clearly then E is not an L„ set. D e f i n i t i o n 6. Let JVCZ+; a subset P of Z is a parallelepiped of dimension N if P has exactly 2A elements and can be represented as a sum. Px+ ...+PN of N two-element subsets. It is easy to see that if E does not contain parallelepipeds of arbitrarily large dimension, then E is an L„ set for some n. Thus, an easy consequence of our corollary is that if Bd(E)>0, then E contains parallelepipeds of arbitrarily large dimension; this fact for natural density was first pointed out in [1]. It is also proved in [1] that if E is a /»-Sidon set, a /1(1) set, or a CJC-set, then there is a N£Z+ for which E con-tains no parallelepiped of dimension N; see [1] and [4] for some definitions. It is also known that such an analytically defined E cannot contain arithmetic progressions of S m a l l s u m se ts a n d t h e F a b e r g a p c o n d i t i o n 237 arbitrary length, hence, it also follows from the work of E . SZEMEREDI [ 7 ] that such sets have natural density zero. On the other hand, it is quite easy to construct an L„ set containing arbitrarily long arithmetic progressions. We conclude our paper with two examples which delineate the scope of The-orem 3. E x a m p l e 1. Let {33" + 32 ' : ngZ+}, Aa = {35"+322: n£Z+}U{35"-t-32°: M € Z + } , . . . , A,= {3P?+32'': 73€Z+}U---U{3"." + 32 ' '+ ' w £ Z + } where p, is the t'h t — 1 CO prime and t'=l + Z f - e t S = U A,. Then S is clearly not q2 but since Sc {3s": ti€Z+}+ {32": n€Z+}, 5 is a <?3 set. In addition, it follows from [2, p. 76] that S is an Lx set that is not Fp for any p. An appeal to Theorem 1 now confirms the following assertion: Given any natural number p there is a sequence {«j}JLi s u c h that ° °> | l im (S—nj ) \>p . E x a m p l e 2. Fix g€Z + . Let {«„}, {£>„}, and {<:„} be pairwise disjoint increas-ing sequences of positive integers. Let A\ = {3b>} and A\ = {3r>}. Let A\~ {3b^, 363} and yi2={3c=,3c3}. Let A\= { 3 \ 3\ 36«} and A\= {3cs 3C*, 3C«}, etc. Finally, let n = l By construction, S is Lx (indeed it is Fq+1). Also if {4,} is any sequence such that dB£A"2 for each n, then {3"', 3\ ..., 3"«}c:lim (S-dn). It follows from say [4] that sup {min (\Aj\, \A2\, \A3\): ^ + 4 + is bounded above for all q. Clearly q could have been chosen greater than this bound; thus sup {min (l^xl, \A2\, \A3\): v 4 1 - M 2 + ^ 3 c ' S ' } < s u P IHeq — /iy)| = <jr+l, for q sufficiently large. Examples 1 and 2 show that the hypothesis in Theorem 3 that S not be an L„_2 set cannot be relaxed. References [1] J. J. F . FOURNIER and L. PIGNO, Analytic and arithmetic properties of thin sets, Pacific J. Math., 105 (1983), 115—141. [2] R. E. DRESSLER and L. PIGNO, Some lacunarity conditions for Fourier—Stieltjes transforms, Ark. Math., 13 (1975), 73—77. [3] R . E. DRESSLER, On sets which contain sum sets, Acta Sci. Math., 42 (1980), 233—235. [4] R. E. EDWARDS and K. A. Ross, /j-Sidon sets, J. Fund. Anal., 15 (1974), 404—427. [5] G. W. JOHNSON and G. S. WOODWARD, On p-Sidon sets, Indiana J. Math., 24 (1974), 161—168. [6] A. Y. KHINCHIN, Three pearls of number theory, Graylock Press (Rochester, N. Y., 1952). [7] E. SZEMEREDI, On sets of integers containing no k elements in arithmetic progression, Acta Arith., 27 (1975), 199—245. DEPARTMENT O F MATHEMATICS KANSAS STATE UNIVERSITY MANHATTAN, KANSAS 66 506, U.S.A. 

Small sum sets and the Faber gap condition

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Small sum sets and the Faber gap condition

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