aa r X i v : . [ m a t h . C O ] F e b Digital almost nets
Boris Bukh ∗ Ting-Wei Chao ∗ Abstract
Digital nets (in base 2) are the subsets of [0 , d that contain the expected number of pointsin every not-too-small dyadic box. We construct sets that contain almost the expected numberof points in every such box, but which are exponentially smaller than the digital nets. We alsoestablish a lower bound on the size of such almost nets. We call a subinterval of [0 , basic (in base q ) if it is of the form (cid:2) aq k , a +1 q k (cid:1) . A basic box is a productof basic intervals, i.e., a set of the form Q di =1 (cid:2) a i q ki , a i +1 q ki (cid:1) . If q = 2, a basic interval is called a dyadicinterval, and a basic box is called a dyadic box.We say that a set P ⊂ [0 , d is a ( m, ε ) -almost net in base q if it is of size | P | = q n m for somenatural number n and (1 − ε ) m ≤ | β ∩ P | ≤ (1 + ε ) m for every basic box β of volume vol( β ) = q − n .The case ε = 0 has been well studied. If ε = 0, almost nets are known as ( t, m, s ) -nets in theliterature. They are used extensively in discrepancy theory and numerical integration algorithms,and are subject to numerous works, including a book devoted exclusively to them [6]. It is known[9, Theorem 3] that, for each d , there exist arbitrarily large ( m, m ≤ q d . On theother hand, m must grow exponentially with d for large enough nets [7] (see also [8] for asymptoticanalysis of the bound in [7]).In contrast to these results, for ε >
0, we construct ( m, ε )-almost nets with m being only polyno-mial in d . Theorem 1.
For any prime q , any d ≥ , and any positive integers m, n satisfying m ≥ d log( dq ) ,there exists a set P ⊂ [0 , d of size mq n such that, for any basic box β of volume q − n , − r d log( dq ) m ! m ≤ | β ∩ P | ≤ r d log( dq ) m ! m. In particular, for every < ε < / and every d ≥ , there exist arbitrarily large ( m, ε ) -almost netsin base q with m ≤ ε − d log( dq ) . ∗ Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA. Supported in partby U.S. taxpayers through NSF CAREER grant DMS-1555149. Email: [email protected] , [email protected] R d without convex holes of size 4 d + o ( d ) rather than 2 d that can be obtained from the conventional nets. See [5] for details.The construction behind Theorem 1 is a minor modification on the construction in [4]. Whereasthe construction in [4] uses primes in Z , this construction uses irreducible polynomials in F q [ x ]. Thereason for this change is to make the denominators be powers of the same prime q . Furthermore,because addition in F q [ x ] satisfies ultrametric inequality, and because we do not need to worry aboutboxes that are not basic, several details in the new construction are simpler. As such, we do not makeany claims about the novelty. Our purpose in writing the present note is to record the details of theconstruction for its application to convex holes. We also hope that almost nets will find applicationsin many other areas that currently use the conventional nets.We do not know when the bound in Theorem 1 is sharp. The following is the best lower boundwe were able to prove. Its dependence on ε is close to optimal, as long as ε is not too small, butthe dependence on d is poor. In the special case ε = 0, we recover the lower bound m = Ω( s ) in( t, m, s )-nets via a proof different than those in [7, 8]. Theorem 2.
Assume that there exists an ( m, ε ) -net P ⊆ [0 , d in base q , then the following holds.If ε ≥ / √ d , then m ≥ Ω (cid:0) log dq ε log(1 /ε ) (cid:1) . If / √ d ≥ ε ≥ e − d/ , then m ≥ Ω( q − k − ε − ) , where k = /ε )log d − log log(1 /ε ) .In particular, if ε = ω ( d − t ) for some constant t , then we have m ≥ Ω q,t (1 /ε ) .If ε = o ( e − cd ) for some constant c such that < c < min(1 / , /q ) , then we get an exponentiallower bound m ≥ Ω (cid:0) q − e c ′ d (cid:1) , where c ′ = 2 c (1 − log q/ log(1 /c )) . Open problem.
It would be interesting to prove a result similar to Theorem 1 which applies to allboxes, not only to basic boxes.
Acknowledgment.
We are thankful to Ron Holzman for useful discussions.
Let t = ⌈ q d + 2 ⌉ . Since the number of irreducible polynomials of degree t in F q [ x ] is1 t ( X i | t µ ( i ) q t/i ) ≥ t ( q t − q t/ ) ≥ d, we may pick d distinct irreducible polynomials p , . . . , p d of degree t in F q [ x ]. We associate each ofthese d polynomials to the respective coordinate direction. We will be interested in canonical boxes ,which are the boxes of the form B = d Y i =1 (cid:20) a i q k i t , a i + 1 q k i t (cid:19) . ≤ a i < q k i t , i = 1 , , . . . , d .We say that a polynomial f ∈ Z [ x ] is a basic polynomial if deg f < t and all of its coefficients arein { , , . . . , q − } .For an irreducible polynomial p ∈ F q [ x ] of degree t and a polynomial f ∈ F q [ x ], we define the base- p expansion of f to be f = f + f p + · · · + f ℓ p ℓ , where each f i is a basic polynomial. Put r p ( f ) def = ( f (1 /q )+ f (1 /q ) q − t + · · · + f ℓ (1 /q ) q − ℓt ) /q , where we view the basic polynomials f , f , . . . , f ℓ as polynomial functions on R . Note that r p ( f ) ∈ [0 , r : F q [ x ] → [0 , d by r ( f ) def = (cid:0) r p ( f ) , . . . , r p d ( f ) (cid:1) . Definition 3.
We say that a box β is good if β is a basic box of volume vol( β ) = q − n . Let B be thesmallest canonical box containing β . We call ( B, β ) a good pair . Note that vol( B ) ≤ q − n + dt − .Suppose B is a canonical box. Write it as B = Q i (cid:2) a i /q k i t , ( a i + 1) /q k i t (cid:1) , and consider r − ( B ).The set r − ( B ) consists of the solutions to the system f ≡ a ′ (mod p k ) ,f ≡ a ′ (mod p k ) , ... f ≡ a ′ d (mod p k d d ) , where a ′ i = f i, + f i, p i + · · · + f i,k i − p k i − i and f i, , f i, , . . . , f i,k i − are the unique basic polynomialssatisfying a i = ( f i, (1 /q ) + f i, (1 /q ) q − t + . . . + f i,k i − (1 /q ) q − ( k i − t ) /q .By the Chinese Remainder theorem, the set r − ( B ) is of the form A ( B ) + D ( B ) F q [ x ] where D ( B ) def = p k p k · · · p k d d and A ( B ) is the unique element in r − ( B ) of degree less than t ( k + . . . + k d ).Note that deg D ( B ) = t ( k + . . . + k d ) = − log q (vol( B )).Given a good pair ( B, β ), define L B ( β ) def = { g ∈ F q [ x ] : r (cid:0) A ( B ) + gD ( B ) (cid:1) ∈ β } . Claim 1.
The set L def = { L B ( β ) : ( B, β ) is a good pair } is of size at most q dt .Proof. Let (
B, β ) be a good pair. Write B and β in the form B = d Y i =1 (cid:20) a i q k i t , a i + 1 q k i t (cid:19) , β = d Y i =1 (cid:20) a i q k i t + b i q ( k i +1) t , a i q k i t + c i q ( k i +1) t (cid:19) . The condition r (cid:0) A ( B ) + gD ( B ) (cid:1) ∈ β is equivalent to A ( B ) + gD ( B ) ∈ a ′ + p k J (mod p k +11 ) ,A ( B ) + gD ( B ) ∈ a ′ + p k J (mod p k +12 ) , ... A ( B ) + gD ( B ) ∈ a ′ d + p k d d J d (mod p k d +1 d ) , J i consist of the basic polynomials f such that f (1 /q ) /q ∈ [ b i , c i ).On the other hand, A ( B ) + gD ( B ) ≡ a ′ + ( α + gδ ) p k (mod p k +11 ) ,A ( B ) + gD ( B ) ≡ a ′ + ( α + gδ ) p k (mod p k +12 ) , ... A ( B ) + gD ( B ) ≡ a ′ d + ( α d + gδ d ) p k d d (mod p k d +1 d )for some α i , δ i ∈ F q [ x ] / ( p i ) , i = 1 , , . . . , d . There are at most q dt different choices for ( α i , δ i ) di =1 .Also, there are at most q dt different choices for ( b i , c i ) di =1 satisfying 0 ≤ b i < c i ≤ q t . Since L B ( β ) isdetermined by ( α i , δ i , b i , c i ) di =1 , the claim is true.To each canonical box B of volume between q − n and q − n + dt − inclusive we assign a type , so thatboxes of the same type behave similarly. Formally, let A ( B ) be the polynomial obtained from thepolynomial A ( B ) by setting the coefficients of 1 , x, x , . . . , x n − dt − to zero. Similarly, let D ( B ) be thepolynomial obtained from D ( B ) by setting the coefficients of 1 , x, x , . . . , x n − dt − to zero. The typeof B is then the pair T ( B ) def = (cid:0) A ( B ) , D ( B ) (cid:1) .Note that, from q − n ≤ vol( B ) ≤ q − n + dt − and deg D ( B ) = − log q (vol( B )) it follows that n − dt + 1 ≤ deg D ( B ) ≤ n. (1) Claim 2.
The number of types is at most q dt .Proof. Since deg A ( B ) < deg D ( B ) ≤ n , only the dt (resp. 2 dt ) leading coefficients of A ( B ) (resp. D ( B ))may be non-zero. Hence, the number of types is at most q dt × q dt = q dt .For a type T = ( A , D ), let Y ( T ) def = {A + g D : g ∈ F q [ x ] } . Note that if T = T ( B ), then Y ( T )is an approximation to r − ( B ). That is to say, the respective elements of Y ( T ) and of r − ( B ) differonly in low-degree coefficients.Let I k denote polynomials of degree less than k in F q [ x ]. Our construction will be a union of setsof the form h + I n − dt where deg h ≤ n + dt .We first prove that there is no difference between the behaviors of Y ( T ) and r − ( B ) intersecting h + I n − dt are the same. Claim 3.
Suppose T ( B ) = (cid:0) A ( B ) , D ( B ) (cid:1) . Then for any polynomial h ∈ I n + dt and any polynomial g , A ( B ) + g D ( B ) ∈ h + I n − dt if and only if A ( B ) + gD ( B ) ∈ h + I n − dt .Proof. If A ( B ) + g D ( B ) ∈ h + I n − dt , then deg( A ( B ) + g D ( B )) < n + dt . Since deg A ( B ) < n anddeg D ( B ) ≥ n − dt ,it follows that deg g < dt . From the definition of A ( B ) and D ( B ), the coefficientsof x n − dt , x n − dt +1 , . . . in A ( B ) + g D ( B ) are the same as the respective coefficients in A ( B ) + gD ( B ).The opposite direction is similar.For a type T and L ∈ L that satisfy T = T ( B ) and L = L B ( β ) for some good pair ( B, β ), define Y T ( L ) def = {A + g D : g ∈ L } . Y T ( L ) is the approximation to r − ( β ) induced by the approximation Y ( B )to r − ( B ). Claim 4.
The set Y T ( L ) def = Y T ( L ) ∩ I n + dt is of size exactly q dt .Proof. Let (
B, β ) be a good pair such that T = T ( B ) and L = L B ( β ). From the previous claim,we know that the size of Y T ( L ) is the same as the size of r − ( β ) ∩ I n + dt . By the Chinese remaindertheorem, each of the canonical boxes of volume q − ( ⌊ n/t ⌋ + d ) t contains equally many points from r ( I n + dt ).Since n ≤ ( ⌊ n/t ⌋ + d ) t , the number of points in β ∩ r ( I n + dt ) is equal to q n + dt vol( β ) = q dt . Claim 5.
Let h be chosen uniformly from I n + dt . Then |Y T ( L ) ∩ ( h + I n − dt ) | is with probability q − dt and is otherwise.Proof. Let q ∈ Y T ( L ) be arbitrary. Clearly Pr[ q ∈ ( h + I n − dt )] = q − dt . The events of the form q ∈ h + I n − dt are mutually disjoint as q ranges over Y T ( L ). Indeed, suppose T = ( A , D ) and q, q ′ ∈ Y T ( L ) are such that q, q ′ ∈ h + I n − dt for some h ∈ I n + dt . We may write q = A + g D and q ′ = A + g ′ D . Then q − q ′ = ( g − g ′ ) D ∈ I n − dt . Since deg D ( B ) ≥ n − dt , this implies that g = g ′ andhence q = q ′ .In the combination with Claim 4, this implies thatPr (cid:2) |Y T ( L ) ∩ ( h + I n − dt ) | = 1 (cid:3) = q − dt q dt = q − dt . Sample q dt m elements uniformly at random from I n + dt , independently from one another. Let H be the resulting multiset, and consider the multiset H + I n − dt def = { h + f : h ∈ H, f ∈ I n − dt } . Fora type T and L ∈ L that satisfy T = T ( B ) and L = L B ( β ) for some good pair ( B, β ), define therandom variable N T ,L def = |Y T ( L ) ∩ ( H + I n − dt ) | . This random variable is distributed according to thebinomial distribution Binom( q dt m, q − dt ).Let ε = p dt log q/m . Note that ε < p d (2 log d + 3 log q ) /m < p d log( dq ) /m , and inparticular ε < /
2. Hence, ε / − ε / ≥ ε /
4. By the tail bounds for the binomial distribution [3,Theorems A.1.11 and A.1.13] we obtainPr (cid:2) N T ,L − m > εm (cid:3) < e − ( ε / − ε / m < q − dt / , Pr (cid:2) N T ,L − m < − εm (cid:3) < e − ε m/ < q − dt / . From Claims 1 and 2 and the union bound it then follows that there exists a choice of H suchthat N T ,L is bounded between (1 − ε ) m and (1 + ε ) m whenever T = T ( B ), L = L B ( β ) and ( B, β ) isa good pair. By Claim 3, this implies that the number of points in any good box β of volume q − n ,the size β ∩ r ( H + I n − dt ) is bounded between (1 − ε ) m and (1 + ε ) m .Hence the multiset r ( H + I n − dt ) in [0 , d is of size exactly mq n and satisfies the conclusion of thetheorem. To obtain a set satisfying the same conclusion, we may perturb the points of r ( H + I n − dt )slightly to ensure distinctness. 5 Proof of Theorem 2
We shall derive Theorem 2 from the following lemma.
Lemma 4.
For any positive integers n, d , prime q , and positive real numbers m, ε with n ≥ d ≥ ,and ε < / , if there exists an ( m, ε ) -almost net P ⊆ [0 , d in base q of size q n m , then m ≥ Ω (cid:0) log( (cid:0) dk (cid:1) ) q k ε log(1 /ε ) (cid:1) , for any integer k such that ≤ k ≤ d/ and ε ≥ (cid:18) dk (cid:19) − / (2) holds.Proof. Let B be the box [0 , /q n − k ) × [0 , d − . For any point v = ( v , . . . , v d ) ∈ B , write itscoordinates in base q as v i = (0 .v i, v i, . . . ) q . Noting that the first n − k digits of v are zero, we let X ( v ) be the first non-trivial digit of v , i.e., X ( v ) def = v ,n − k +1 . Similarly, let X i ( v ) def = v i, for i ≥ X , . . . , X d for a randomly chosenpoint of B . However, we do not directly appeal to the known bound on the size of probability spacessupporting almost independent random variables (see e.g. [1, 2]) because those bounds are formulatedfor { , } -valued random variables, whereas X , . . . , X d take q distinct values.Let S def = P ∩B , and t def = | S | . Since P is an ( m, ε )-almost net, it follows that t is between q k (1 − ε ) m and q k (1 + ε ) m . Assume v , . . . , v t are all the points in S .For x ∈ R , let e q ( x ) def = exp(2 πx/q ). Let U be a t -by- (cid:0) dk (cid:1) matrix, where the rows are indexed by [ t ]and the columns are indexed by (cid:0) [ d ] k (cid:1) . The general entry of U is U i,J def = e q (cid:0)X j ∈ J X j ( v i ) (cid:1) . Also, define A def = t U U ∗ . Claim 6.
The diagonal terms in A are all . The off-diagonal terms are, in absolute value, boundedabove by ε .Proof. The general term of A is given by A J ,J = t t X i =1 e q (cid:16) X j ∈ J X j ( v i ) − X j ∈ J X j ( v i ) (cid:17) . If J = J , this is clearly 1.Suppose J = J . Note that, for any choice of α = ( α j ) j ∈ J ∆ J , the set { v ∈ B : X j ( v ) = α j for j ∈ J ∆ J }
6s a basic box of volume q − n +2 k −| J ∆ J | . Thus, for any τ ∈ [ q ], the region B τ def = n v ∈ B : X j ∈ J X j ( v ) − X j ∈ J X j ( v ) ≡ τ (mod q ) o can be partitioned into q | J ∆ J |− many basic boxes of volume q − n +2 k −| J ∆ J | each. Since we have − n + 2 k − | J ∆ J | ≥ − n , it follows that the number of i such that v i ∈ b α is bounded between q k − (1 − ε ) m and q k − (1 + ε ) m . Thus, | A J ,J | = 1 t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q X τ =1 | B τ ∩ S | e q ( τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q X τ =1 q k − me q ( τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 1 t q X τ =1 | εq k − e q ( τ ) | = εq k mt ≤ ε. We apply [1, Theorem 2.1] to the matrix ( A + ¯ A ) /
2. We obtain that, if (cid:0) dk (cid:1) − / ≤ ε < / , then2 q k (1 + ε ) m ≥ A ) ≥ rank (cid:0) ( A + ¯ A ) / (cid:1) ≥ Ω( log( ( dk ) ) ε log(1 /ε ) ). Therefore, m ≥ Ω (cid:0) log( (cid:0) dk (cid:1) ) q k ε log(1 /ε ) (cid:1) . The right hand side of lemma 4 is a decreasing function of k for k ∈ [1 , d/ k as small as possible. If ε ≥ / √ d , then we may set k = 1 and get m ≥ Ω (cid:0) log dq ε log(1 /ε ) (cid:1) . If 1 / √ d ≥ ε ≥ e − d/ , then we may set k = /ε )log d − log log(1 /ε ) . From the assumption on k , we have k ≤ log(1 /ε ). Therefore, (cid:18) dk (cid:19) ≥ ( d/k ) k ≥ exp (cid:0) /ε )log d − log log(1 /ε ) (log d − log k ) (cid:1) ≥ exp (cid:0) /ε )log d − log log(1 /ε ) (log d − log log(1 /ε )) (cid:1) ≥ ε , and so (2) holds. Hence, we may apply Lemma 4 with ⌈ k ⌉ in place of k and obtain m ≥ Ω( q − k − ε − ) . In particular, if ε = ω ( d − t ) for some constant t , then k is also a constant, and so m ≥ Ω q,t (1 /ε ) inthis case.If ε = o ( e − cd ) for some constant c such that 0 < c < min(1 / , /q ), then the ( m, ε )-net is alsoan ( m, e − cd )-net, when d is large enough. We may apply the result above with e − cd in place of ε . In this case, the calculations above yield k = 2 cd/ log(1 /c ), and we get m ≥ Ω (cid:0) q − e c ′ d (cid:1) where c ′ = 2 c (cid:0) − log q/ log(1 /c ) (cid:1) . 7 eferences [1] Noga Alon. Perturbed identity matrices have high rank: proof and applications. Combinatorics,Probability and Computing , 18(1–2):3—-15, 2009.[2] Noga Alon, Alexandr Andoni, Tali Kaufman, Kevin Matulef, Ronitt Rubinfeld, and Ning Xie.Testing k -wise and almost k -wise independence. In STOC’07—Proceedings of the 39th AnnualACM Symposium on Theory of Computing , pages 496–505. ACM, New York, 2007.[3] Noga Alon and Joel H. Spencer.
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