AA remark on the minimal dispersion
A. E. Litvak
Abstract
We improve known upper bounds for the minimal dispersion ofa point set in the unit cube and its inverse in both the periodic andnon-periodic settings. Some of our bounds are sharp up to logarithmicfactors.
AMS 2010 Classification: primary: 52B55, 52A23; secondary: 68Q25, 65Y20.
Keywords: complexity, dispersion, largest empty box, torus
In this note we deal with the minimal dispersion of a point set in the unit cube.The dispersion of a point set in the d -dimensional unit cube [0 , d is defined as themaximal volume of an axis parallel box in the cube which does not contain anypoint from the set. Then the minimal dispersion is a function of two variables, n and d , which minimizes the dispersion over all possible choices of n points. Suchdefinition was introduced in [9] modifying a notion from [7]. Due to importantapplications and due to the fact that the problem is very interesting by itself, ithas attracted a considerable attention in recent years. We refer to [1, 3, 5, 10, 11,13, 15, 16] and references therein for the history of the problem and its relationto other branches as well as for the best known bounds (see also [8, 12, 14] forthe dispersion of certain sets). We improve known upper bounds for the minimaldispersion and its inverse function. We will also consider the minimal dispersionon the torus and discuss the sharpness of our results. We would like to emphasizethat we look at the dispersion as at a function of two variables, without trying tofix one of the variables. Instead, we consider both variables growing to infinity andour bounds depend on the relations between variables. The main novelty in ourproof is a better construction of a family of axis parallel boxes (periodic or non-periodic) needed to be checked for a random choice of points. It seems that our a r X i v : . [ m a t h . C A ] J u l onstruction also leads to better bounds for recently introduced in [6] k -dispersion(where, given set of n points, one allows axis parallel boxes to have inside at most k points from this set), but we do not pursue this direction. We start with notations. Given a measurable set A ⊂ R d , we denote its d -dimensional volume by | A | . We also use the same notation | M | for the cardinalityof a finite set M (it always will be clear from the context what | · | means). By R d we denote the set of all axis parallel boxes contained in the cube Q d := [0 , d ,that is R d := (cid:40) d (cid:89) i =1 I i | I i = [ a i , b i ) ⊂ [0 , (cid:41) . The dispersion of a finite set P ⊂ Q d is defined asdisp( P ) = sup {| B | | B ∈ R d , B ∩ P = ∅} . Then the minimal dispersion is defined as the function of two variables — thecardinality of a set of points P ⊂ Q d and the dimension, namelydisp ∗ ( n, d ) = sup | P | = n disp( P ) . We also define its inverse as N ( ε, d ) = min { n ∈ N | disp ∗ ( n, d ) ≤ ε } . Since in our proofs we use a random choice of points, it will be natural to proveresults in terms of the function N ( ε, d ) and then to provide the corresponding(equivalent) bounds for the minimal dispersion itself. First we discuss the known bounds. In [1] it was shown that for ε < / − ε ) log d ε ≤ N ( ε, d ) ≤ d +1 ε , (1)where the upper bound is due to Larcher, improving the previous bound via pri-morials due to Rote and Tichy [9] (see also [3]) and the lower bound is the firstnon-trivial bound showing that the minimal dispersion grows with the dimension.Note that one trivially has disp ∗ ( n, d ) ≥ / ( n + 1), hence N ( ε, d ) ≥ /ε − lthough estimates in (1) are tight when the dimension d is small and ε goesto 0, there is a huge gap between the upper and lower bounds when the dimensionstarts to grow to infinity. Using random choice of points uniformly distributed in Q d , Rudolf [10] obtained N ( ε, d ) ≤ dε log (cid:18) ε (cid:19) (2)(this bound with different numerical constants also follows from much more generalresults in [2], where the VC dimension of R d was used, and from the fact that thisVC dimension equals to 2 d ). Estimate (2) is better than the upper bound in (1)in the regime ε ≥ exp( − C d ) , where C > ε is not extremelysmall with respect to the dimension, the gap in bounds is polynomial in d andlogarithmical in 1 /ε . Another important feature of the Rudolf proof is that arandom choice of points uniformly distributed on Q d gives the result.It was natural to conjecture that N ( ε, d ) behaves as d/ε , especially in view ofcorresponding bounds in the periodic setting (see below), however, surprisingly,Sosnovec [11] was able to improve the upper bound for ε < / N ( ε, d ) ≤ C ε log d, where the order of magnitude of C ε was essentially (1 /ε ) (1 /ε ) . This dependencewas significantly improved in [15] by Ullrich and Vyb´ıral, who showed that C ε = 2 log dε (cid:18) log (cid:18) ε (cid:19)(cid:19) works. They also conjectured that N ( ε, d ) behaves as log d/ε . The Sosnovec–Ullrich–Vyb´ıral upper bound is better in the regime ε ≥ C (log d ) d . The Sosnovec–Ullrich–Vyb´ıral proof is also based on a random choice of points,but instead of the uniform distribution on Q d they use uniform distribution ona certain lattice, gaining in the case of large ε . We discuss this in more detailsbelow. Let us also mention that in the same paper Sosnovec proved that the unction N ( ε, d ) completely changes the behaviour at ε = 1 /
4, more precisely, heproved that for every ε > / N ( ε, d ) ≤ (cid:22) ε − / (cid:23) . Thus, for ε > /
4, the function N ( ε, d ) is not growing with d . Note that clearly N (1 / , d ) = 1 (by taking the point (1 / , / , ..., / ε ≤ / N ( ε, d ) ≤ C ln dε ln (cid:0) ε (cid:1) , if ε ≥ ln dd , C dε ln (cid:0) ε (cid:1) , if ln dd ≥ ε ≥ exp( − C d ) , C d ε , if ε ≤ exp( − C d ) , where 1 < C < In this note we improve the known bounds in the regime ε ≥ exp( − C d ). Our firstresult improves bounds when ε is not large. Theorem 1.1.
There exists an absolute constant C ≥ such that the followingholds. Let d ≥ and ε ∈ (0 , / . Then ( i ) N ( ε, d ) ≤ C ln dε ln (cid:18) ε (cid:19) , provided that ε ≤ exp( − d ) , ( ii ) N ( ε, d ) ≤ C dε ln ln (cid:18) ε (cid:19) , provided that ε ≥ exp( − d ) . Moreover, the random choice of points with respect to the uniform distribution onthe cube Q d gives the result with high probability. We would like to emphasize that if ε ≤ exp( − d ) then, in view of (1), Theo-rem 1.1 yields ln d ε ≤ N ( ε, d ) ≤ C ln dε ln (cid:18) ε (cid:19) , thus the gap in bounds is only logarithmical in 1 /ε . In the second case the im-provement is only in substitution of ln(1 /ε ) with ln ln(1 /ε ) comparing to Rudolf’sbound.Our proof is also based on a random choice of points. A standard way touse randomness is to show that a certain “good” event E holds with a non-zero robability. Equivalently, one needs to show that the complement of E , the event E c , holds with small probability. In order to do that, one tries to cover E c bycertain events, called individual events, to obtain good bounds on probabilitiesof individual events, and then to use the union bound. In this scheme one needsto have a good balance between (small) probabilities of individual events and the(large but not too large) size of the covering set. Since we need to prove thatthere exists a set P of n points such that there is no rectangles of volume ε without a point from P , the natural idea would be to construct a finite set N ofrectangles having reasonably large volume and such that property “each rectanglein N contains a point from P ” implies the property “each rectangle in R d ofvolume at least ε contains a point from P .” In the case of uniform distribution onthe cube Q d , that is, in the case when the set P consists of N points independentlydrawn from the uniform distribution, an individual bound, that is, a bound on theevent that a given box B ∈ N contains a point from P , is simply given by thevolume of B , therefore the main difficulty is to construct the set N of not toolarge cardinality. Rudolf used the concept of δ -cover [10, 4] to construct N and toestimate its size. We introduce the notion of δ -net (see Definition 2.1), which fitsbetter for random procedure described above and allows to obtain better boundson its size, see Propositions 3.1 and 3.4.As usual in probabilistic proofs, we obtain the result with high probability.Very recently, Hinrichs, Krieg, Kunsch, and Rudolf [5] investigated the best boundthat one can get using a random choice of points and showed that one cannot expectanything better than max (cid:26) cε ln (cid:18) ε (cid:19) , d ε (cid:27) , (3)where c > d in the first estimate and up toln ln(1 /ε ) in the second estimate).In the case of large ε we can improve the bound. The next theorem providesbetter bounds in the regime ε ≥ (ln d ) / ( d ln ln(2 d )). Theorem 1.2.
There exists an absolute constant C ≥ such that the followingholds. Let d ≥ and ε ∈ (0 , / be such that ε ≥ ln dd . Then N ( ε, d ) ≤ C ln dε ln (cid:18) ε (cid:19) . This improves the Ullrich–Vyb´ıral bound by removing one ln(1 /ε ) factor. Theproof of this theorem also uses random points uniformly distributed on the cube Q d ,however, as Hinrichs–Krieg–Kunsch–Rudolf’s result shows, one cannot expect abound better than d/ε , therefore one needs to adjust the distribution of the points. ne way to adjust randomness was suggested by Sosnovec and then improved byUllrich and Vyb´ıral. They substituted the uniform distribution on the cube by auniform distribution on a certain lattice inside the cube. This led to the logarithmicin d upper bound (by the price of an additional factor 1 /ε ). Careful analysis oftheir proofs in comparison with Rudolf’s proof shows that the main advantageof the use of a lattice is that the points on the lattice are ε -separated from theboundary of the cube. This leads to our adjustment of the uniform distribution onthe cube — if a uniformly distributed over the cube random point falls too closeto the boundary we slightly shift it to the interior, to ensure that it is ε -separatedfrom the boundary. In the next section we introduce the function φ ε , which servesthis purpose. Unfortunately, the size of δ -nets is still too large, to deal with large ε ,so we additionally introduce the notion of dinets — nets in the sense of dispersion (see Definition 2.2), which allows us to reduce the cardinality of a covering set (seeProposition 3.6) and hence to apply the union bound.The upper bonds for ε ≤ / N ( ε, d ) ≤ C ln dε ln (cid:0) ε (cid:1) , if ε ≥ ln dd ln ln(2 d ) , C dε ln ln (cid:0) ε (cid:1) , if ln dd ln ln(2 d ) ≥ ε ≥ e − d , C ln dε ln (cid:0) ε (cid:1) , if e − d ≤ ε ≤ exp( − C d ) , C d ε , if ε ≤ exp( − C d )or in the following picture showing the corresponding regions. z = d d C ln 1ln C d Cd ln 1ln C d dC z e = d z e = ln ln(2 )ln d dz d = n terms of the minimal dispersion, Theorems 1.1 and 1.2 are equivalent to thefollowing theorem. Theorem 1.3.
There exists an absolute constant C ≥ such that the followingholds. Let d ≥ and n ≥ d . Then ( i ) disp ∗ ( n, d ) ≤ C ln dn ln (cid:16) n ln d (cid:17) , provided that n ≥ e d d ln d, ( ii ) disp ∗ ( n, d ) ≤ C dn ln ln (cid:16) nd (cid:17) , provided that d ln ln d ln d ≤ n ≤ e d d ln d, ( iii ) disp ∗ ( n, d ) ≤ (cid:18) C ln dn ln (cid:16) n ln d (cid:17)(cid:19) / , provided that n ≤ d ln ln d ln d . Moreover, in the first two cases the random choice of points with respect to theunifrom distribution on the cube Q d gives the result with high probability. The corresponding dispersion on the torus can be described in terms of periodicaxis parallel boxes. We denote such a set by (cid:101) R d , that is (cid:101) R d := (cid:40) d (cid:89) i =1 I i ( a, b ) | a, b ∈ Q d (cid:41) , where I i ( a, b ) := (cid:40) ( a i , b i ) , whenever 0 ≤ a i < b i ≤ , [0 , \ [ b i , a i ] , whenever 0 ≤ b i < a i ≤ . The dispersion of a finite set P ⊂ Q d on the torus, the minimal dispersion on thetorus, and its inverse are defined in the same way as above, but using sets from (cid:101) R d , that is (cid:103) disp( T ) = sup {| B | | B ∈ (cid:101) R d , B ∩ T = ∅} , (cid:103) disp ∗ ( n, d ) = sup | P | = n (cid:103) disp( P ) , and (cid:101) N ( ε, d ) = min { n ∈ N | (cid:103) disp ∗ ( n, d ) ≤ ε } . It is known that dε ≤ (cid:101) N ( ε, d ) ≤ dε (cid:18) ln d + ln (cid:18) ε (cid:19)(cid:19) , where the lower bound was proved by Ulrich [13] and the upper bound is due toRudolf [10] (since there are no good bounds on the VC dimension of (cid:101) R d , results f [2] are not directly applicable here). We would like to emphasize that contraryto the non-periodic case, even in the case of large ε , the lower bound is at least d .We improve the Rudolf upper bound in the case ε ≤ /d . Theorem 1.4.
There exists an absolute constant C ≥ such that the followingholds. Let d ≥ and ε ∈ (0 , / . Then ( i ) (cid:101) N ( ε, d ) ≤ C ln dε ln (cid:18) ε (cid:19) , provided that ε ≤ exp( − d ) , ( ii ) (cid:101) N ( ε, d ) ≤ C d ln dε , provided that ε ≥ exp( − d ) . Moreover, the random choice of points with respect to the uniform distribution onthe cube Q d gives the result with high probability. Equivalently, for d ≥ and n ≥ d ln d we have ( i ) disp ∗ ( n, d ) ≤ C ln dn ln (cid:16) n ln d (cid:17) , provided that n ≥ e d d ln d, ( ii ) disp ∗ ( n, d ) ≤ C d ln dn , provided that d ln d ≤ n ≤ e d d ln d. Our bound on (cid:101) N ( ε, d ) reduces the factor d in Rudolf’s estimate to ln d in thecase when ε ≤ exp( − d ) and removes the summand ln(1 /ε ) if exp( − d ) < ε ≤ /d .However, if ε ≥ /d , it gives the same order ( d ln d ) /ε .The proof is the same as for Theorem 1.1, using random points and a δ -netconstructed for periodic boxes. Unfortunately, in the construction of nets for thesecond bound in Theorem 1.1 and for the bound in Theorem 1.2, we essentially usethat boxes are not periodic and therefore the construction cannot be extended tothe periodic case (for Theorem 1.2 it is also clear in view of the Ullrich lower boundon (cid:101) N ( ε, d )). We would also like to note that the Hinrichs–Krieg–Kunsch–Rudolf’sresult on best possible lower bound (3) which may be obtained by using randompoints uniformly distributed on the cube holds for the periodic setting as well,therefore the factor ln(1 /ε ) in our first estimate is unavoidable by this method. Inthe second case, ε ≥ exp( − d ), we have ln(1 /ε ) ≤ d , so there is a hope to removeln d factor and to obtain the best possible estimate, on the other hand it is possiblethat the bound is the best possible for this method. We need more notations. Given a positive integer m we denote [ m ] = { , , ..., m } .Given ε >
0, we consider sets of (periodic) axis parallel of volume at least ε , B ε,d := (cid:110) B ∈ R d | | B | ≥ ε (cid:111) and (cid:101) B ε,d := (cid:110) B ∈ (cid:101) R d | | B | ≥ ε (cid:111) . e introduce the following definition. Definition 2.1 ( δ -net for B ε,d ) . Given ε, δ > we say that N ⊂ R d is a δ -net for B ε,d if for every B ∈ B ε,d there exists B ∈ N such that B ⊂ B and | B | ≥ (1 − δ ) | B | . We define a δ -net for (cid:101) B ε,d in a similar way. To deal with large ε with respect to the dimension, say when ε ≥ /d , weadjust the definition of a δ -net by introducing the notion of δ -dinet — a δ -net in asense of dispersion . The key idea leading to this approach is an observation thatwe do not need to consider points which are too close to the boundary of the cube Q d . As we mentioned in the introduction, this idea was already implicitly used in[11, 15]. First given ε ∈ (0 , /
2) define an auxiliary function φ ε : [0 , → [ ε, − ε ]by φ ε ( t ) = ε if 0 ≤ t < ε,t if ε ≤ t ≤ − ε, − ε if 1 − ε < t ≤ . Given x ∈ Q d we also write φ ε ( x ) for { φ ε ( x i ) } di =1 . Definition 2.2 ( δ -dinet for B ε,d ) . Given ε, δ > we say that N ⊂ R d is a δ -dinetfor B ε,d if for every B ∈ B ε,d there exists B ∈ N such that | B | ≥ (1 − δ ) | B | and such that for every x ∈ Q d the following implication holds x ∈ B = ⇒ φ ε ( x ) ∈ B Note that every δ -dinet N for B ε,d has the following property allowing to boundfrom above the number of points needed to have a given dispersion, namely, forevery n ≥
1, every set of points P = { x , ..., x n } ⊂ Q d the statement “each boxfrom N contains at least one point from P ” implies the statement “each box from B ε,d contains at least one point from φ ε ( P ) ” . A variant of the following lemma using random points and the unionbound was proved in [10] (see Theorem 1 there). We provide a proof forcompleteness.
Lemma 2.3.
Let d ≥ and ε, δ ∈ (0 , . Let N be either a δ -net for B ε,d ora δ -dinet for B ε,d and let (cid:101) N be a δ -net for (cid:101) B ε,d . Assume both |N | ≥ and | (cid:101) N | ≥ . Then N ( ε, d ) ≤ |N | (1 − δ ) ε and (cid:101) N ( ε, d ) ≤ | (cid:101) N | (1 − δ ) ε . emark 2.4. As usual for proofs involving the union bound, our proof showsthat the random choice of points gives the result with high probability, moreprecisely with probability at least − / |N | . In the case of δ -nets the random-ness is with respect to the independent uniform choice of points on Q d , whilein the case of δ -dinets one needs to adjust the choice independent uniformlydistributed points by the function φ ε .Proof. We show a proof for a δ -net N for B ε,d , the other two cases are thesame. Let N be a δ -net for B ε,d . Consider N independent random points X ,..., X N uniformly chosen from Q d . By the definition of a δ -net, it is enoughto show that for every B ∈ N with | B | ≥ v := (1 − δ ) ε there exists j ≤ N such that X j ∈ B . Fix such a box B . Using that the volume of B is at least v and the independence of X j ’s, we obtain P ( {∀ j ≤ N : X j / ∈ B } ) = (1 − v ) N < exp( − vN ) . Therefore, by the union bound, P ( {∃ B ∈ N : | B | ≥ v and ∀ j ≤ N : X j / ∈ B } ) < |N | exp( − vN ) . Thus, as far as |N | exp( − vN ) ≤
1, there exists a realization of X j ’s with thedesired property. This implies both Lemma 2.3 and Remark 2.4. As is seen from Lemma 2.3 and Remark 2.4, to prove our theorems, it isenough to construct nets of not so large cardinality. The next simple obser-vation is one of key ideas in our estimates. Let ε > (cid:96) , ..., (cid:96) d > d (cid:89) i =1 (cid:96) i ≥ ε. Denote by σ = σ ( (cid:96) , ..., (cid:96) d ) a permutation such that (cid:96) σ (1) ≤ (cid:96) σ (2) ≤ ... ≤ (cid:96) σ ( n ) (4)(for each sequence we fix one such permutation). Then for every j ≥ (cid:96) σ ( j ) ≥ (cid:32) j (cid:89) i =1 (cid:96) σ ( i ) (cid:33) /j ≥ ε /j > − ln(1 /ε ) j . (5)10 naive approach to approximate rectangles from B ε,d is to say that givena rectangle B = (cid:81) di =1 I i ∈ B ε,d the smallest length (cid:96) i = | I i | is at least ε .Therefore, we can take (1 / (4 ε ))-net M in [0 ,
1] and approximate each I i with segments having endpoints in M . This approach would lead to a netof the order (1 / (4 ε )) d , which is not acceptable for our purpose (this wouldalso lead to a huge loss in volume, but already the size of a net is too large).Instead, we use formula (5), to say that the larger i the coarser net in [0 ,
1] isneeded in order to approximate the corresponding interval I σ ( i ) . Of course,simultaneously, we need to control the loss in volume in our approximation.The next proposition utilizes this idea. It works for both the periodic andnon-periodic settings. Since we will be using this result in several dimensions,it would be convenient to formulate it for boxes in R m . Proposition 3.1.
Let m ≥ be an integer and ε ∈ (0 , . There are (1 / -nets N and (cid:101) N for B ε,m and (cid:101) B ε,m respectively, each of them of cardinality atmost (14 m ) m ε (2 m ) . Remark 3.2. If m = 2 k for some integer k then our proof gives slightlybetter estimate, namely (24 m ) m ε m . Remark 3.3.
Clearly, Lemma 2.3 and Remark 2.4 combined with this propo-sition (applied with m = d ) yield Theorem 1.4 as well as the first bound inTheorem 1.1.Proof. The construction of nets in B ε,d and (cid:101) B ε,m are essentially the same. Weprovide a proof for a net in (cid:101) B ε,m , since the proof for a net in B ε,m is somewhateasier — we do not need to consider intervals I i ( a, b ) with a i > b i .Fix k ≥ k ≤ m < k +1 . Fix a partition of [ m ] into k + 1disjoint sets A , ..., A k +1 with | A | = 2, | A k +1 | = m − k (this set is emptyif m = 2 k ), and | A j | = 2 j − for 2 ≤ j ≤ k . For j ≤ k + 1 denote δ ( j ) = 2 − k − ε − j and D j = { , δ ( j ) , δ ( j ) , ..., s j δ ( j ) } , where s j = (cid:98) /δ ( j ) (cid:99) (note that dealing with B ε,m we do not need to have 0 in D j ). 11e are now ready to define a part of our net corresponding to this par-tition of [ m ] as the set N ∗ ( A , ..., A k +1 ) := (cid:110) m (cid:89) i =1 I i ( x, y ) | x, y ∈ Q m , ∀ j ≤ k +1 ∀ i ∈ A j : x i (cid:54) = y i ∈ D j (cid:111) . Then the cardinality of this set can be estimated as |N ∗ ( A , ..., A k +1 ) | ≤ k +1 (cid:89) j =1 (cid:89) i ∈ A j | D j | ( | D j | − ≤ k +1 (cid:89) j =1 (cid:89) i ∈ A j δ ( j ) ) ≤ m k +1 (cid:89) j =1 (cid:89) i ∈ A j k +3 ε − j ≤ m (cid:18) k +3 ε (cid:19) k +1 (cid:89) j =2 (cid:18) k +3 ε − j (cid:19) j − = 2 m k +1 ( k +3) ε k +1) ≤ m (64 m ) m ε (2 m ) (note that if m = 2 k , then the set A k +1 is empty and j runs between 1 and k , which leads to the bound from Remark 3.2).To complete the construction, we take the union over all partitions of [ m ]into such sets A , ..., A k +1 , N := (cid:91) N ∗ ( A , ..., A k +1 ) . The number of partitions can be estimated as (cid:18) m k (cid:19)(cid:18) k k − (cid:19)(cid:18) k − k − (cid:19) ... (cid:18) (cid:19) ≤ m , hence |N | ≤ m (8 m ) m ε (2 m ) ≤ (14 m ) m ε (2 m ) . It remains to show that N is indeed a (1 / (cid:101) B ε,m . Let a, b ∈ Q m and B = (cid:81) mi =1 I i ( a, b ) be of volume at least ε . For i ≤ m let (cid:96) i be the lengthof I i ( a, b ). Let σ = σ ( (cid:96) , ..., (cid:96) m ) be the permutation defined by (4). Considerthe following partitions of [ m ], A σ = σ ( { , } ) , A σk = σ ( { k + 1 , ..., m } ) , and A σj = σ ( { j − + 1 , ..., j } ) , ≤ j ≤ k , and note that by (5) for every j ≤ k + 1 and every i ∈ A σj one has I i ( a, b ) = (cid:96) i ≥ (cid:96) σ (2 j − ) ≥ ε / j − = 2 k +3 δ ( j ) . (6)12ake a box B = (cid:81) mi =1 I i ( x, y ) from N ∗ ( A σ , ..., A σk +1 ) such that for every j ≤ k + 1 and every i ∈ A σj one has a i ≤ x i , b i ≥ y i , x i − a i ≤ δ ( j ) , and b i − y i ≤ δ ( j ) (if a i > s j δ ( j ) we take x i = 0). The lower bound (6) on the length of I i ( a, b )implies that I i ( x, y ) ⊂ I i ( a, b ). Thus, B ⊂ B and, using (6) again, | B | = k +1 (cid:89) j =1 (cid:89) i ∈ A σj | I i ( x, y ) | ≥ k +1 (cid:89) j =1 (cid:89) i ∈ A σj (cid:0) (cid:96) i − δ ( j ) (cid:1) = m (cid:89) i =1 (cid:96) i k +1 (cid:89) j =1 (cid:89) i ∈ A σj (cid:18) − δ ( j ) (cid:96) i (cid:19) ≥ | B | k +1 (cid:89) j =1 (cid:18) − δ ( j ) (cid:96) σ (2 j − ) (cid:19) | A σj | ≥ | B | k +1 (cid:89) j =1 (cid:18) − k +2 (cid:19) | A σj | ≥ | B | (cid:18) − k +2 (cid:19) m ≥ | B | (cid:18) − m (cid:19) m ≥ | B | . This completes the proof.Next we show how to improve the bound of Proposition 3.1 for non-periodic boxes in the case when ε is not very small with respect to dimen-sion, say, when 4 ln(1 /ε ) ≤ d . The key observation here is that in the case4 ln(1 /ε ) ≤ d a rectangle B = (cid:81) di =1 I i ∈ B ε,d has many intervals I i of lengthclose to one, namely, by (5), | I σ ( i ) | ≥ − /L whenever i ≥ L ln(1 /ε ). Forsuch an interval we do not need to take a net in [0 ,
1] in order to approximatethe end points — it is enough to approximate the left end point by a netin [0 , /L ] and the right end point by a net in [1 − /L, Proposition 3.4.
Let d ≥ be an integer, ε ∈ (0 , / and assume that d ≥ /ε ) . Then B ε,m admits a (3 / -net of cardinality at most exp ( Cd ln ln(1 /ε )) , where C ≥ is an absolute constant. Remark 3.5.
Clearly, Lemma 2.3 and Remark 2.4 combined with this propo-sition yield the second bound in Theorem 1.1. roof. The proof is similar to the proof of Proposition 3.1, but we deal morecarefully with the approximation of long segments.Set k to be the smallest integer such that 2 k ≥ /ε ) and let m = 2 k .Clearly, k ≥ m ≥
2. Then d ≥ /ε ) > m . Fix an integer n ≥ k suchthat 2 n ≤ d < n +1 . Fix a partition of [ d ] into n − k + 2 disjoint sets A , ..., A n − k +1 with | A | = m , | A n − k +1 | = d − n (this set is empty if d = 2 n ), and | A j | = 2 k + j − for 1 ≤ j ≤ n − k . Denote δ = 18 d , D = { δ, δ, ..., sδ } , and D = { − δ, − δ, ..., − sδ } , where s j = (cid:98) /δ (cid:99) .Next, for every 1 ≤ j ≤ n − k + 1 we consider the set P j ⊂ D × D ofall pairs ( p, q ) satisfying p ∈ D , q ∈ D , p < q , and p ≤ − k − j ln(1 /ε ) + δ and q ≥ − − k − j ln(1 /ε ) − δ. Using 2 k ≥ /ε ) and δ = 1 / (8 d ), we observe that the cardinality of P j is | P j | ≤ (cid:18) − k − j ln(1 /ε ) δ + 1 (cid:19) ≤ (cid:18) d j + 1 (cid:19) ≤ d j − Let N ( A ) be the (1 / n := (24 m ) m ε m for B ε,m from Proposition 3.1 constructed in R A (see also Remark 3.2). Let N ∗ = N ∗ ( A , ..., A n − k +1 ) be the set of all boxes (cid:81) di =1 [ x i , y i ) such that (cid:89) i ∈ A [ x i , y i ) ∈ N ( A )and for every 1 ≤ j ≤ n − k + 1 and for every i ∈ A j the pair ( x i , y i ) ∈ P j .Then, using 2 n ≤ d < n +1 and m = 2 k , the cardinality of N ∗ can beestimated as |N ∗ | ≤ |N ( A ) | n − k +1 (cid:89) j =1 (cid:89) i ∈ A j | P j | ≤ n n − k +1 (cid:89) j =1 (cid:18) d j − (cid:19) | A j | ≤ n n − k +1 (cid:89) j =1 d k + j ( j − k + j − ≤ n d n +2 ( n − k − n +1 = n (cid:18) k +4 d n +1 (cid:19) n +1 ≤ n (cid:0) m (cid:1) d = (24 m ) m ε m (16 m ) d . d ] into suchsets A , ..., A n − k +1 , N := (cid:91) N ∗ ( A , ..., A n − k +1 ) . The number of partitions can be estimated as (cid:18) d n (cid:19)(cid:18) n n − (cid:19)(cid:18) n − n − (cid:19) ... (cid:18) k +1 k (cid:19) ≤ d − k +1 ≤ d − m , hence |N | ≤ d − m (24 m ) m ε m (16 m ) d ≤ (12 m ) m ε m (24 m ) d ≤ (24 m ) d ε m . Using that m ≤ /ε ) ≤ d , we obtain |N | ≤ exp (6 d ln(24 m ) + 2(log m )(ln(1 /ε )) ≤ exp ( Cd ln ln(1 /ε )) , where C ≥ N is indeed a (1 / B ε,d . Let a, b ∈ Q d with a i < b i for all i ≤ d , and B = (cid:81) di =1 [ a i , b i ) be of volume at least ε . For i ≤ d let (cid:96) i = b i − a i . Let σ = σ ( (cid:96) , ..., (cid:96) d ) be the permutation defined by (4).Consider the following partitions of [ d ], A σ = σ ([ m ]) , A σk = σ ( { n +1 , ..., d } ) , and A σj = σ ( { k + j − +1 , ..., k + j } ) , ≤ j ≤ n − k .Fix for a moment 1 ≤ j ≤ n − k + 1 and i ∈ A σj . Using 2 k ≥ /ε )and (5), we observe that b i − a i = (cid:96) i ≥ (cid:96) σ (2 k + j − ) > − ln(1 /ε )2 k + j − ≥ − − j . (7)Take a pair ( x i , y i ) ∈ D × D satisfying a i ≤ x i , b i ≥ y i , x i − a i ≤ δ, and b i − y i ≤ δ. Then y i − x i ≥ b i − a i − δ > − − j − δ > y i ≥ b i − δ > − ln(1 /ε )2 k + j − − δ and x i ≤ a i + δ < ln(1 /ε )2 k + j − + δ, in other words the pair ( x i , y i ) ∈ P j . 15onsider the box B = (cid:81) di =1 [ x i , y i ) such that for every 1 ≤ j ≤ n − k + 1and every i ∈ A σj the pair ( x i , y i ) is constructed as above and where B (cid:48) = (cid:89) i ∈ A σ [ x i , y i ) ∈ N ( A σ ) approximates B (cid:48) = (cid:89) i ∈ A σ [ a i , b i )as in Proposition 3.1 (note that m -dimensional volume of B (cid:48) is at least ε , so B (cid:48) ∈ B ε,m ). Then by construction B ∈ N ∗ ( A σ , ..., A σk +1 ), B ⊂ B , and | B (cid:48) | ≥ | B (cid:48) | = 12 (cid:89) i ∈ A σ (cid:96) i . Furthermore, using δ = 1 / (8 d ) and the bound (7) again, | B | = | B (cid:48) | n − k +1 (cid:89) j =1 (cid:89) i ∈ A σj | y i − x i | ≥ (cid:89) i ∈ A σ (cid:96) i n − k +1 (cid:89) j =1 (cid:89) i ∈ A σj ( (cid:96) i − δ )= 12 d (cid:89) i =1 (cid:96) i n − k +1 (cid:89) j =1 (cid:89) i ∈ A σj (cid:18) − δ(cid:96) i (cid:19) = 12 | B | n − k +1 (cid:89) j =1 (cid:18) − δ − − j (cid:19) | A σj | ≥ | B | (cid:18) − d (cid:19) d − n n − k (cid:89) j =1 (cid:18) − d (cid:19) k + j − ≥ | B | (cid:18) − d (cid:19) d ≥ | B | . This completes the proof.Finally, we want to improve bounds in the case of large ε . The fol-lowing proposition is an almost immediate consequence of Proposition 3.1and definitions. The key observation here is also the fact that a rectangle B = (cid:81) di =1 I i ∈ B ε,d has many intervals I i of the length close to one, but nowthey will be so close to one, that we can substitute them just by [0 , φ ε , if the length of I i is at least 1 − ε then forevery z ∈ [0 ,
1] one has φ ε ( z ) ∈ I i , hence we do not need to approximatesuch intervals. This leads to our definition of a dinet and to better boundsof cardinality of dinets versus regular nets. Unfortunately, this also leads toan additional factor 1 /ε in the final bound. As in the previous proposition,such an approach essentially uses that we are in the non-periodic setting. Proposition 3.6.
Let d ≥ be an integer, ε ∈ (0 , / and assume that d ≥ (ln(1 /ε )) /ε . There is a (1 / -dinet N for B ε,m of cardinality at most exp (cid:18) /ε ) ln(18 d ) ε (cid:19) . emark 3.7. Clearly, Lemma 2.3 combined with this proposition yields The-orem 1.2. We can also use Remark 2.4 to claim that a random choice ofpoints works with high probability, but here the randomness will be with re-spect to the uniform distribution on the cube adjusted by the function φ ε . Proof.
Fix the smallest integer m ≥ (ln(1 /ε )) /ε . Given subset A ⊂ [ d ] ofcardinality m , let N ( A ) be the (1 / n := (14 m ) m ε (2 m ) for B ε,m from Proposition 3.1 constructed in R A . Let N ∗ ( A ) be the set of allboxes (cid:81) di =1 [ x i , y i ) such that (cid:89) i ∈ A [ x i , y i ) ∈ N ( A )and for every i / ∈ A , [ x i , y i ) = [0 , N = (cid:91) A ⊂ [ d ] | A | = m N ( A ) . Then the cardinality of N is at most (cid:18) dm (cid:19) n ≤ (cid:18) edm (cid:19) m n ≤ (14 em d ) m ε (2 m ) ≤ (18 d ) m ε (2 m ) . Since (ln(1 /ε )) /ε ≤ m ≤ d and m ≤ /ε )) /ε , this implies |N | ≤ exp(4 m ln(18 d ) + 2 log (2 m ) ln(1 /ε )) ≤ exp (cid:18) /ε ) ln(18 d ) ε (cid:19) . Now we show that N is a (1 / B ε,d . Let a, b ∈ Q d with a i < b i for all i ≤ d , and B = (cid:81) di =1 [ a i , b i ) be of volume at least ε . For i ≤ n let (cid:96) i = b i − a i . Let σ = σ ( (cid:96) , ..., (cid:96) d ) be the permutation defined by (4) and denote A σ = σ ([ m ]). Consider the box B = (cid:81) di =1 [ x i , y i ) such that [ x i , y i ) = [0 , i / ∈ A σ and B (cid:48) = (cid:89) i ∈ A σ [ x i , y i ) ∈ N ( A σ ) approximates B (cid:48) = (cid:89) i ∈ A σ [ a i , b i )17s in Proposition 3.1 (note that m -dimensional volume of B (cid:48) is at least ε , so B (cid:48) ∈ B ε,m ). Then by construction B ∈ N ∗ ( A σ ), and | B | = | B (cid:48) | ≥ | B (cid:48) | = 12 (cid:89) i ∈ A σ (cid:96) i ≥ | B | . Finally assume that z ∈ B . If i / ∈ A σ then using (5) and m ≥ (ln(1 /ε )) /ε we have b i − a i = (cid:96) i ≥ (cid:96) σ ( m ) > − ε. Therefore, φ ε ( z i ) ∈ [ ε, − ε ] ⊂ [ a i , b i ). Assume i ∈ A σ . Note that in this case z i ∈ [ x i , y i ) ⊂ [ a i , b i ) , and b i − a i ≥ ε (otherwise | B | < ε ). If ε ≤ z i ≤ − ε then φ ε ( z i ) = z i hence φ ε ( z i ) ∈ [ a i , b i ) . If 0 ≤ z i < ε then the interval [ a i , b i ) contains a point smallerthan ε and has length at least ε . Then it must contain ε = φ ε ( z i ). Similarly,if 1 − ε < z i < a i , b i ) must contain 1 − ε = φ ε ( z i ). This proves thatif z ∈ B then φ ε ( z ) ∈ B . Thus, N is a (1 / B ε,d . This completesthe proof. Acknowledgments
The author was introduced to this problem during the 2017 MFO workshop “Perspectives in High-dimensional Probability and Convexity.”
The authoris grateful to MFO, to the organizers, and participants of the workshop. Theauthor is also grateful to A. Zelnikov for his help with the picture.
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