Near-Optimal Lower Bounds for ɛ-nets for Half-spaces and Low Complexity Set Systems

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1 Near-Optimal Lower Bouds for -ets for Half-spaces ad Low Compleity Set Systems Adrey Kupavskii Nabil H. Mustafa Jáos Pach Abstract Followig groudbreakig work by Haussler ad Welzl 1987, the use of small -ets has become a stadard techique for solvig algorithmic ad etremal problems i geometry ad learig theory. Two sigificat recet developmets are: i a upper boud o the size of the smallest -ets for set systems, as a fuctio of their so-called shallow-cell compleity Cha, Grat, Köema, ad Sharpe; ad ii the costructio of a set system whose members ca be obtaied by itersectig a poit set i R 4 by a family of half-spaces such that the size of ay -et for them is Ω 1 log 1 Pach ad Tardos. The preset paper completes both of these aveues of research. We i give a lower boud, matchig the result of Cha et al., ad ii geeralize the costructio of Pach ad Tardos to half-spaces i R d, for ay d 4, to show that the geeral upper boud, O d log 1, of Haussler ad Welzl for the size of the smallest -ets is tight. 1 Itroductio Let X be a fiite set ad let R be a system of subsets of a uderlyig set cotaiig X. I computatioal geometry, the pair X, R is usually called a rage space. A subset X X is called a -et for X, R if X R for every R R with R X X. The use of small-sized -ets i geometrically defied rage spaces has become a stadard techique i discrete ad computatioal geometry, with may combiatorial ad algorithmic cosequeces. I most applicatios, -ets precisely ad provably capture the most importat quatitative ad qualitative properties that oe would epect from a radom sample. Typical applicatios iclude the eistece of spaig trees ad simplicial partitios with low crossig umber, upper bouds for discrepacy of set systems, LP roudig, rage searchig, streamig algorithms; see [13, 18]. For ay subset Y X, defie the projectio of R o Y to be the set system R Y := { Y R : R R }. The Vapik-Chervoekis dimesio or, i short, the VC-dimesio of the rage space X, R is the miimum iteger d such that R Y < 2 R for ay subset Y X with Y > d. Accordig to the Sauer Shelah lemma [21, 23] discovered earlier by Vapik ad Chervoekis [24], for ay rage space X, R whose VC-dimesio is at most d ad for ay subset Y X, we have R Y d i=0 Y i = O Y d. A straightforward samplig argumet shows that every rage space X, R has a -et of size O 1 log R X. The remarkable result of Haussler ad Welzl [10], based o the previous work of Vapik ad Chervoekis [24], shows that much smaller -ets eist if we assume that our rage space has small VC-dimesio. Haussler ad Welzl [10] showed that if the VC-dimesio of a rage space X, R is at most d, the by pickig a radom sample of size Θ d log d, we obtai a -et with positive probability. Actually, they oly used the weaker assumptio that R Y = O Y d for every Y X. This boud was later improved to 1 + o1 d log 1, as 1 ad d is large [11]. I the sequel, we will refer to this result as the -et theorem. The key feature of the -et theorem is that it guaratees the eistece of a Moscow Istitute of Physics ad Techology, G-SCOP laboratory, Greoble. kupavskii@yade.ru. The work of Adrey Kupavskii has bee supported i part by the Swiss Natioal Sciece Foudatio Grats ad ad by the grat N of the Russia Foudatio for Basic Research. Uiversité Paris-Est, LIGM, Equipe A3SI, ESIEE Paris, Frace. mustafa@esiee.fr. The work of Nabil H. Mustafa has bee supported by the grat ANR SAGA JCJC-14-CE EPFL, Lausae ad Réyi Istitute, Budapest. pach@cims.yu.edu. The work of Jáos Pach has bee partially supported by Swiss Natioal Sciece Foudatio Grats ad

2 -et whose size is idepedet of both X ad R X. Furthermore, if oe oly requires the VC-dimesio of X, R to be bouded by d, the this boud caot be improved. It was show i [11] that give ay > 0 ad iteger d 2, there eist rage spaces with VC-dimesio at most d, ad for which ay -et must have size at least 1 2 d + 1 dd+2 + o1 d log 1. The effectiveess of -et theory i geometry derives from the fact that most geometrically defied rage spaces X, R arisig i applicatios have bouded VC-dimesio ad, hece, satisfy the precoditios of the -et theorem. There are two importat types of geometric set systems, both ivolvig poits ad geometric objects i R d, that are used i such applicatios. Let R be a family of possibly ubouded geometric objects i R d, such as the family of all half-spaces, all balls, all polytopes with a bouded umber of facets, or all semialgebraic sets of bouded compleity, i.e., subsets of R d defied by at most D polyomial equatios or iequalities i the d variables, each of degree at most D. Give a fiite set of poits X R d, we defie the primal rage space X, R as the set system iduced by cotaimet i the objects from R. Formally, it is a set system with the set of elemets X ad sets {X R : R R}. The combiatorial properties of this rage space deped o the projectio R X. Usig this termiology, Rado s theorem [13] implies that the primal rage space o a groud set X, iduced by cotaimet i half-spaces i R d, has VC-dimesio at most d + 1 [18]. Thus, by the -et theorem, this rage space has a -et of size O d log 1. I may applicatios, it is atural to cosider the dual rage space, i which the roles of the poits ad rages are swapped. As above, let R be a family of geometric objects rages i R d. Give a fiite set of objects S R, the dual rage space iduced by them is defied as the set system hypergraph o the groud set S, cosistig of the sets S := {S S : S} for all R d. It ca be show that if for ay X R d the VC-dimesio of the rage space X, R is less tha d, the the VC-dimesio of the dual rage space iduced by ay subset of R is less tha 2 d [13]. Recet progress. I may geometric scearios, however, oe ca fid smaller -ets tha those whose eistece is guarateed by the -et theorem. It has bee kow for a log time that this is the case, e.g., for primal set systems iduced by cotaimet i balls i R 2 ad half-spaces i R 2 ad R 3. Over the past two decades, a umber of specialized techiques have bee developed to show the eistece of small-sized -ets for such set systems [3, 4, 5, 6, 7, 8, 11, 12, 14, 16, 20, 25, 26]. Based o these successes, it was geerally believed that i most geometric scearios oe should be able to substatially stregthe the -et theorem, ad obtai perhaps eve a O 1 upper boud for the size of the smallest -ets. I this directio, there have bee two sigificat recet developmets: oe positive ad oe egative. Upper bouds. Followig the work of Clarkso ad Varadaraja [8], it has bee gradually realized that if oe replaces the coditio that the rage space X, R has bouded VC-dimesio by a more refied combiatorial property, oe ca prove the eistece of -ets of size o 1 log 1. To formulate this property, we eed to itroduce some termiology. Give a fuctio ϕ : N R +, we say that the rage space X, R has shallow-cell compleity ϕ if there eists a costat c = cr > 0 such that, for every Y X ad for every positive iteger l, the umber of at most l-elemet sets i R Y is O Y ϕ Y l c. Note that if the VC-dimesio of X, R is d, the for every Y X, the umber of elemets of the projectio of the set system R to Y satisfies R Y = O Y d. However, the coditio that X, R has shallow-cell compleity ϕ for some fuctio ϕ = O d, 0 < d < d 1 ad some costat c = cr, implies ot oly that R Y = O Y 1+d +c, but it reveals some otrivial fier details about the distributio of the sizes of the smaller members of R Y. Several of the rage spaces metioed earlier tured out to have low shallow-cell compleity. For istace, the primal rage spaces iduced by cotaimet of poits i disks i R 2 or half-spaces i R 3 have shallow-cell compleity ϕ = O1. I geeral, it is kow [13] that the primal rage space iduced by cotaimet of poits by half-spaces i R d has shallow-cell compleity ϕ = O d/2 1. Defie the uio compleity of a family of objects R, as the maimum umber of faces boudary pieces of all dimesios that the uio of ay members of R ca have; see [1]. Applyig a simple probabilistic techique developed by Clarkso ad Shor [9], oe ca fid a iterestig relatioship betwee the uio compleity of a family of objects R ad the shallow-cell compleities of the dual rage spaces iduced by subsets S R. Suppose that the uio compleity of a family R of objects i the plae is O ϕ, for some well-behaved o-decreasig fuctio ϕ. The the umber of at most l-elemet subsets i the dual rage space iduced by ay S R is O l 2 S S l ϕ l = O S ϕ S l [22]; i.e., the dual rage space iduced by S has shallow-cell compleity O ϕ. Accordig to the above 2

3 defiitios, this meas that for ay S R ad for ay positive iteger l, the umber of subsets S S l for which there is a poit p R 2 cotaied i all elemets of S, but i oe of the elemets of S \ S, is at most O S ϕ S l. For small values of l, the poits p are ot heavily covered. Thus, the correspodig cells S S S \ T S\S T of the arragemet S are shallow, ad the umber of these shallow cells is bouded from above. This eplais the use of the term shallow-cell compleity. A series of elegat results [20, 3, 6, 26] illustrate that if the shallow-cell compleity of a set system is ϕ = o, the it permits smaller -ets tha what is guarateed by the -et theorem. The followig theorem represets the curret state of the art; see [15] for a simple proof of this statemet. Theorem A. Let X, R be a rage space with shallow-cell compleity ϕ, where ϕ = O d for some costat d. The, for every > 0, it has a -et of size O 1 log ϕ 1, where the costat hidde i the O-otatio depeds o d. Proof. Sketch. The mai result i [6] shows the eistece of -ets of size O 1 log ϕ X for ay o-decreasig fuctio ϕ 1. To get a boud idepedet of X, first compute a small /2-approimatio A X for X, R [13]. It is kow that there is such a A with A = O d 2 log 1 = O 1 3, ad for ay R R, we have R A A R X 2. I particular, ay R R with R X cotais at least a 2 -fractio of the elemets of A. Therefore, a /2-et for A, R A is a -et for X, R. Computig a /2-et for A, R A gives the required set of size O 2 log ϕ A = O 1 log ϕ 1 3 = O 1 log ϕ 1. Note that i the bouds o the sizes of -ets based o VC-dimesio, we eplicitly state the depedece o d. O the other had, i the bouds based o shallow-cell compleity, we will assume that d is a costat. Lower bouds. It was cojectured for a log time [14] that most geometrically defied rage spaces of bouded Vapik-Chervoekis dimesio have liear-sized -ets, i.e., -ets of size O 1. These hopes were shattered by Alo [2], who established a superliear but barely superliear! lower boud o the size of -ets for the primal rage space iduced by straight lies i the plae. Shortly after, Pach ad Tardos [19] maaged to establish a tight lower boud of Ω 1 log 1 for the size of -ets i primal rage spaces iduced by half-spaces i R 4, ad i several other geometric scearios. Theorem B. [19] Let F deote the family of half-spaces i R 4. For ay > 0 ad ay sufficietly large iteger, there eists a set X R 4 of poits such that i the primal rage spaces X, F, the size of every -et is at least 1 9 log 1. Our cotributios. The aim of this paper is to complete both aveues of research opeed by Theorems A ad B. Our first theorem, proved i Sectio 2, geeralizes Theorem B to R d, for d 4. It provides a asymptotically tight boud i terms of both ε ad d, ad hece completely settles the -et problem for half-spaces. Theorem 1. For ay iteger d 4, real > 0 ad ay sufficietly large iteger 0, there eist primal rage spaces X, F iduced by -elemet poit sets X ad collectios of half-spaces F i R d such that the size of every -et for X, F is at least d/4 9 log 1. As was metioed i the first subsectio, for ay d 1, the VC-dimesio of ay rage space iduced by poits ad half-spaces i R d is at most d + 1. Thus, Theorem 1 matches, up to a costat factor idepedet of d ad, the upper boud implied by the -et theorem of Haussler ad Welzl. Noga Alo poited out to us that it is very easy to show that for a fied > 0, the lower boud for -ets i rage spaces iduced by half-spaces i R d has to grow at least liearly i d. To see this, suppose that we wat to obtai a 1 3-et, say, for the rage space iduced by ope half-spaces o a set X of 3d poits i geeral positio i R d. Notice that for this we eed at least d + 1 poits. Ideed, ay d poits of X spa a hyperplae, ad oe of the ope half-spaces determied by this hyperplae cotais at least X 3 poits. The key elemet of the proof of Theorem B [19] was to costruct a set B of k + 32 k 2 ais-parallel rectagles i the plae such that for ay subset of them there is a set Q of at most 2 k 1 poits that hit oe of the rectagles that belog to this subset ad all the rectagles i its complemet the precise statemet is give i Sectio 3. We geeralize this statemet to R d by costructig roughly d 2 times more ais-parallel boes 2 tha i the plaar case, but the size of the set Q remais the same size. I Sectio 3, we prove 1 Their result is i fact for the more geeral problem of small weight -ets. 2 A ais-parallel bo i R d is the Cartesia product of d itervals. For simplicity, i the sequel, they will be called boes. 3

4 Lemma 2. Let k, d 2 be itegers. The there eists a set B of d 2 k + 32k 2 ais-parallel boes i R d such that for ay subset S B, oe ca fid a 2 k 1 -elemet set Q of poits with the property that i Q B for ay B B \ S, ad ii Q B = for ay B S. I the et sectio we show how this lemma implies the boud of Theorem 1, which is d 4 times better tha the boud i Theorem B. The proof of Lemma 2 will be give i Sectio 3. I Sectio 4, we show that the boud i Theorem A caot be improved. Defiitio 1. A fuctio ϕ : R + R + is called submultiplicative if there eists a 0 R + such that for every α, 0 < α < 1, ad > 0, we have ϕ α 1/α ϕ. Some eamples of submultiplicative fuctios are c for ay positive c, 2 log, log, log log, log, ad the iverse Ackerma fuctio. Theorem 3. Let d be a fied positive iteger ad let ϕ : R + R +, ϕ, be a mootoically icreasig submultiplicative fuctio that teds to ifiity such that ϕ = O d. The, for ay > 0, there eist rage spaces X, F that have i shallow-cell compleity ϕ, ad for which ii the size of ay -et is Ω 1 log ϕ 1. Theorem 3 becomes iterestig whe ϕ = o ad the upper boud O 1 log ϕ 1 i Theorem A improves o the geeral upper boud O 1 log 1 guarateed by the -et theorem. Theorem 3 shows that, eve if ϕ = o, this improved boud is asymptotically tight. The best upper ad lower bouds for the size of small -ets i rage spaces with a give shallow-cell compleity ϕ are based o purely combiatorial argumets, ad they imply directly or idirectly all kow results o -ets i geometrically defied rage spaces see [17] for a detailed discussio. This suggests that the itroductio of the otio of shallow-cell compleity provided the right framework for -et theory. 2 Proof of Theorem 1 usig Lemma 2 Let B be a set of d-dimesioal ais-parallel boes i R d. We recall that the dual rage space iduced by B is the set system hypergraph o the groud set B cosistig of the sets B p := {B B : p B} for all p R d. Lemma 4. Let d 1 be a iteger, ad cosider the dual rage space iduced by a set of ais-parallel boes B i R d. The there eists a fuctio f : B R 2d such that for every poit p R d, there is a half-space H i R 2d with {fb : B B p } = H {fb : B B}. Proof. By traslatio, we ca assume that all boes i B lie i the positive orthat of R d. Cosider the fuctio g : B R 2d mappig a bo B = [ l 1, r 1] [ l 2, r 2] [ l d, r d ] to the poit l 1, 1/ r 1, l 2, 1/ r 2,..., l d, 1/r d lyig i the positive orthat of R2d. Furthermore, for ay p = a 1, a 2,..., a d R d i the positive orthat, let C p deote the bo [0, a 1 ] [0, 1/a 1 ] [0, a 2 ] [0, 1/a 2 ] [0, a d ] [0, 1/a d ] i R 2d. Clearly, a poit p lies i a bo B i R d if ad oly if gb C p i R 2d. Thus, g maps the set of boes i B to a set of poits i R 2d, such that for ay poit p i the positive orthat of R d, the set of boes B p B that cotai p are mapped to the set of poits that belog to the bo C p. Note that C p cotais the origi. We complete the proof by applyig the followig simple trasformatio [19, Lemma 2.3] to the set Q = gb: to each poit q Q i the positive orthat of R 2d, we ca assig aother poit q i the positive orthat of R 2d such that for each bo i R 2d that cotais the origi, there is a half-space with the property that q belogs to the bo if ad oly if q belogs to the correspodig half-space. The mappig fb = gb for every B B meets the requiremets of the lemma. Lemma 5. Give ay iteger d 2, a real umber > 0, ad a sufficietly large iteger 0, there eists a set B of ais-parallel boes i R d such that the size of ay -et for the dual set system iduced by B is at least d 2 9 log 1. Proof. Let = α with k N, k 2, ad 1 2 k 1 3 α 2 3. Applyig Lemma 2, we obtai a set B of d 2 k + 32k 2 boes i R d. We claim that the dual rage space iduced by these boes does ot admit a -et of size 1 α B. 4

5 Assume for cotradictio that there is a -et S B with S 1 α B. Accordig to Lemma 2, there eists a set Q of 2 k 1 poits i R d with the property that o bo i S cotais ay poit of Q, but every member of B \ S does. By the pigeohole priciple, there is a poit p Q cotaied i at least B \ S Q α B Q = α B = B 2k 1 members of B \ S. Thus, oe of the at least B members of B hit by p belog to S, cotradictig the assumptio that S was a -et. Hece, the size of ay -et i the dual rage space iduced by B is at least 1 α B = 1 α d 2 k + 32k 2 = 1 αα 2 d 2 k d 2 1 log 1. The system of boes costructed above has a fied umber of elemets, depedig o the value of 1/. We ca obtai arbitrarily large costructios by replacig each bo of B B with several slightly traslated copies of B we refer the reader to [19] for details. Now we are i a positio to establish Theorem 1. By Lemma 4, ay lower boud for the size of -ets i the dual rage space iduced by the set B of boes i R d gives the same lower boud for the size of a -et i the primal rage space o the set of poits fb R 2d correspodig to these boes, i which the rages are half-spaces i R 2d. For ay iteger d 4 ad ay real > 0, Lemma 5 guaratees the eistece of a set B of ais-parallel boes i R d/2 such that ay -et for the dual set system iduced by B has size at least d/2 2 9 log 1 = d/4 9 boud. 3 Proof of Lemma 2 The proof of Lemma 2 is based o the followig key statemet. log 1. This fact, together with Lemma 4, implies the stated Lemma C [19]. Let k 2 be a iteger. The there eists a set R of k + 32 k 2 ais-parallel rectagles i R 2 such that for ay S R, there eists a 2 k 1 -elemet set Q of poits i R 2 with the property that i Q R for ay R R \ S, ad ii Q R = for ay R S. Deote the - ad y-coordiates of a poit p R 2 by p ad yp respectively, ad set m = d 2. Let R = {R 1,..., R t }, t = k + 32 k 2, be a set of rectagles satisfyig the coditios of Lemma C. By scalig, oe ca assume that R [0, 1] 2 for every R R. Give that a bo i R d is the product of d itervals, the idea of the costructio is to lift the rectagles i Lemma C, i.e., the set R, to boes i R d. So a rectagle R R ca be mapped to a bo i R d which is the product d itervals: the first two beig the itervals defiig R, ad the other d 2 itervals i the product beig the full iterval [0, 1]. Oe ca the agai lift the same set R i a o-iterferig way by mappig R to a bo whose 3-rd ad 4-th itervals are the itervals of R ad the remaiig itervals are [0, 1]. I this way, by packig itervals of each R R ito disjoit coordiates, oe ca lift R m times to get a set of d 2 R boes i Rd. Formally, for i = 1... m, defie the ijective fuctios f i that map a poit i R 2 to a product of d itervals i R d, as follows. f i p = [0, 1] [0, 1] p yp [0, 1] [0, 1], p R 2. }{{}}{{} 2i 2 itervals d 2i itervals This mappig lifts each rectagle R R to the bo f i R = {f i p : p R}, ad each set of rectagles R R to the set of boes f i R = {f i R : R R }. We ow show that B = m i=1 f ir is the desired set of d 2 k + 32k 2 boes i R d. Let S B be a fied set of boes. For ay ide i [1, m], set R i R to be the set of preimage rectagles uder f i of the boes i S f i R, i.e., R i satisfies S f i R = f i R i. Let Q i = {q1, i..., q i 2 } R 2 be the set of k 1 poits hittig all rectagles i R \ R i ad o rectagle i R i ; such a set eists by Lemma C. Now we argue that the set { q 1 j, yqj 1,..., qj m, yqj m : j [1, 2 ]} k 1 if d is eve, Q = { q 1 j, yqj 1,..., qj m, yqj m, 1 : j [1, 2 ]} k 1 if d is odd, 5

6 of 2 k 1 poits i R d is the required set for S; i.e., Q hits all the boes i B \ S, ad oe of the boes i B S. Take ay bo B B \ S; the there eists a ide i ad a rectagle R R \ R i such that R is the preimage rectagle of B uder f i. By Lemma C, R cotais a poit q Q i, ad thus B = f i R cotais the poit q Q with q ad yq i its 2i 1-th ad 2i-th coordiates, as all the remaiig itervals defiig B are [0, 1] ad so each such iterval cotais the correspodig coordiate of q. O the other had, let B S be a bo with the preimage rectagle R R i. By Lemma C, R is ot hit by ay poit of Q i, ad thus for ay poit q Q, the 2i 1-th ad 2i-th coordiates caot both be cotaied i the correspodig two itervals defiig B. Therefore, q does ot hit B. 4 Proof of Theorem 3 The goal of this sectio is to establish lower bouds o the sizes of -ets i rage spaces with give shallow-cell compleity ϕ, where ϕ is a submultiplicative fuctio. We will use the followig property of submultiplicative fuctios. Claim 6. Let ϕ : R + R + be a submultiplicative fuctio. The i for all sufficietly large, y R +, we have ϕy ϕϕy, ad ii if there eists a sufficietly large R + ad a costat c such that ϕ c, the ϕ c for every. Proof. Both of these properties follow immediately from the submultiplicativity of ϕ : i. ϕy = ϕy log y ϕy logy y ϕ y log y ϕ y log y y = ϕ ϕy. ii. ϕ log ϕ c = ϕ c log = c log = c. Theorem 3 is a cosequece of the followig more precise statemet. Theorem 7. Let ϕ : R + R + be a mootoically icreasig submultiplicative fuctio which teds to ifiity ad is bouded from above by a polyomial of costat degree. For ay 0 < δ < 1 10, oe ca fid a 0 > 0 with the followig property: for ay 0 < < 0, there eists a rage space with shallow-cell compleity ϕ o a set of = log ϕ 1 elemets, i which the size of ay -et is at least 1 2 δ log ϕ 1. Proof. The parameters of the rage space are as follows: = log ϕ 1, m = = log ϕ 1 ϕ 1 2δ, p =. m Let d be the smallest iteger such that ϕ = O d. By Claim 6, part ii, for ay large eough, we have d 1 ϕ c 1 d, for a suitable costat c 1 1. I the most iterestig case where ϕ = o, we have d = 1. For a small eough, we have c 1 log ϕ 1, so that m = log ϕ 1 log c1 d d log c 1 d log. 1 Cosider a rage space [], F with a groud set [] = {1, 2,..., } ad with a system of m-elemet subsets F, where each m-elemet subset of [] is added to F idepedetly with probability p. The et claim follows by a routie applicatio of the Cheroff boud. Claim 8. With high probability, F 2ϕ 1 2δ. Theorem 7 follows by combiig the et two lemmas that show that, with high probability, the rage space [], F i does ot admit a -et of size less tha 1 2 δ log ϕ 1, ad ii has shallow-cell compleity ϕ. For the proofs, we eed to assume that = δ, d, ϕ is a sufficietly large costat, or, equivaletly, that 0 = 0 δ, d is sufficietly small. Lemma 9. With high probability, the rage space [], F has shallow-cell compleity ϕ. 6

7 The reaso why this lemma holds is that for ay X [] ad ay k, it is very ulikely that the umber of at most k-elemet sets eceeds the umber permitted by the shallow-cell compleity coditio. To boud the probability of the uio of these evets for all X ad k, we simply use the uio boud. Proof. It is eough to show that for all sufficietly large 0, every X [], X =, ad every l m, the umber of sets of size eactly l i F X is Oϕ, as this implies that the umber of sets i F X of size at most l is O ϕl. I the computatios below, we will also assume that l d + 1 2; otherwise if l d, ad assumig 0 2d, we have l d ϕ, d where the last iequality follows by the assumptio o ϕ, provided that is sufficietly large. We distiguish two cases. Case 1: > probability, we have ϕ δ/d. I this case, we trivially upper-boud F X by F. By Claim 8, with high F 2 ϕ 1 2δ 2 ϕ ϕ 1 2δ by Claim 6 2 ϕ ϕ ϕ δ/d 1 2δ as ϕδ/d 1 2δ usig 2 c 1 ϕϕ δ ϕt c1 t d 2c 1ϕ 1 δ 2c 1ϕ 1 δ+δ/d = Oϕ. Case 2: ϕ δ/d. Deote the largest iteger that satisfies this iequality by 1. It is clear that 1 = o recall that ϕ is mootoically icreasig ad teds to ifiity. We also deote the system of all l-elemet subsets of F X by F l X ad the set of all l-elemet subsets of X by X l. Let E be the evet that F does ot have the required ϕ -shallow-cell compleity property. The Pr[E] m l=2 Pr[E l], where E l is the evet that for some X [], X =, there are more tha ϕ elemets i F l X. The, for ay fied l d + 1 2, we have Pr[E l ] 1 = 0 Pr 1 = 0 1 = 0 1 = 0 1 [ ] X [], X =, F l X > ϕ l [ ] Pr For a fied X, X =, {S F X, S = l} = s s= ϕ l s= ϕ l s= ϕ l = 0 s= ϕ 1 l = 0 s= ϕ 1 l = 0 s= ϕ l s Pr [ For a fied X, X =, S ] we have F l X = S X, S = s, l l 1 1 p m l s 1 p m l l s 2 s e e e l s e l e l+1 l 1 l l ϕ p m s s p 3 m l m l s m l 4 e em l 1 e 2 mϕ 1 2δ s 5 ϕ 7

8 I the trasitio to the epressio 3, we used several times i the boud a b ea b b for ay a, b N; ii the iequality 1 p b 1 bp for ay iteger b 1 ad real 0 p 1; ad iii we upper-bouded the last factor of 2 by 1. I the trasitio from 3 to 4 we lower-bouded s by ϕ. We also used the estimate m l m m l, which ca be verified as follows. m l = m l m m l 1 m i m + i + 1 i=0 l m m m m l m l. Fially, to obtai 5, we substituted the formula for p ad used the fact that l l m l = l m l l l l, 2 as 1 = o, m = 4 for < 0 1/4 ad l 2. Deote 2 = 1 δ. We split the epressio 5 ito two sums Σ 1 ad Σ 2. Let Σ 1 := Σ 2 := l = 0 s= ϕ 1 l = 2 s= ϕ e em l 1 e 2 mϕ 1 2δ ϕ e em l 1 e 2 mϕ 1 2δ ϕ s s These two sums will be bouded separately. We have Σ 1 l = 0 s= ϕ l = 0 s= ϕ l = 0 s= ϕ e e e em em s l 1 c 1 2δ 1 e 2 m d 2dδ d 2dδ ϕ 2δ 6 l 1 d+2dδ s Cm d+1 2dδ for some C > 0 δ/2 l 1 d+2dδ s Cm d+1 7 e ϕ l δ 2 2dδ δ2 e 2 l ϕdδ2 2 8 = 0 = 0 2 ϕdδ = o m = 0 = 0 To obtai 6, we used the property that ϕ ϕϕ c 1ϕ d, provided that,, are sufficietly large. To establish 7, we used the fact that 2 = 1 δ ad that em ed log δ/2. I the trasitio to 8, we eeded that l d + 1, d 1 ad that Cm d+1 Cd log d+1 = o δ2 /2. The we lower-bouded s by ϕ. To arrive at 9, we used that l. The last iequality follows from the fact that 0 is large eough, so that ϕ ϕ 0 8/dδ 2 ad that m = o. Net, we tur to boudig Σ 2. First, observe that ϕ 1 2δ ϕ 1 2δ 1 δ 1 δ ϕ 1 2δ 1 δ ϕ 1 δ, where we used the submultiplicativity ad mootoicity of the fuctio ϕ ad the fact that 2 = 1 δ. Secod, ote by that restrictig to be small eough, by the submultiplicativity of ϕ, we have that for ay 2, m log ϕ 1 ϕδ/4 1 ϕδ/3 2 ϕ δ/

9 Substitutig the boud for ϕ 1 2δ i Σ 2, settig C = e 2, ad by 10, we obtai Σ 2 1 l = 2 s= ϕ 1 l = 2 1 = 2 1 = 2 = 1 1 e e em l 1 s Cϕ 2δ/3 em ϕ Cϕ 2δ/3 11 ϕ ϕ e 1+/ϕ m l/ϕ Cϕ 2δ/3 ϕ C ϕ δ/3 ϕ for some costat C > ϕ 2 2ϕ 2 2 2ϕ 2 Cϕ δ/ ϕ 2 2 = o 1 m. I the trasitio to 11, we used that l 2 ad 1 δ 1 /ϕ δ/d 1, which implies that 1 /ϕ δ/2d ad, therefore, em < e log ϕ1/ e log ϕ ϕ δ/2d ϕ δ/2d ϕ δ/3d. To get 12, we used that for some costat c > 1 we have l/ϕ c m/ϕ c log ϕ/ϕ = O1 ad that m ϕ δ/3 for 2 by 10. To obtai 13, observe that 1/2ϕ2 = O1. At the last equatio, we used that 1 = o, e/ 1 as ad that 2 ϕ = Ω 1 δ. We have show that for every l = 2,..., m, we have Pr[E l ] = o1/m, as m teds to ifiity. Thus, we ca coclude that Pr[E] m l=2 Pr[E l] = o1 ad, hece, with high probability the rage space [], F has shallow-cell compleity ϕ.. Now we are i a positio to prove that with high probability the rage space [], F does ot admit a small -et. Lemma 10. With high probability, the size of ay -et of the rage space [], F is at least 1 2 δ log ϕ 1. Proof. Deote by µ the probability that the rage space has a -et of size t = 1 2 δ log ϕ 1 = 1 2 δ. The we have µ Pr [ X is a -et for F ] 1 p t m e p t m 2 e ϕδ/2 = o1. 14 t t X [] X =t To verify the last two iequalities, otice that, sice 1 a > e b for b > a, 0 < < 1 a 1 b, we have t m t p p m m t = ϕ 1 2δ 1 t ϕ 1 2δ 1 m t m δ t t ϕ 1 2δ e mt δ = ϕ 1 2δ e 1 2δ 1+δ log ϕ 1 = ϕ 1 2δ ϕ 1 2δ 1+δ 1 ϕ 1 2δ ϕ 1 2δ 1+δ ϕ δ/2. 13 Here the last but oe iequality follows from 1 holds, because δ 1/10. ad from the mootoicity of ϕ. The last iequality Thus, Lemma 9 ad Lemma 10 imply that with high probability the rage space [], F has shallow-cell compleity ϕ ad it admits o -et of size less tha 1 2 δ log ϕ 1. This completes the proof of the theorem. 9

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