The Span of Random Binary Matrices and the Lemma of Littlewood and

In this section, we pass from the setting of deterministic binary N ×n matrices to Bernoullirandom matrices. Similarly as in the previous section, they induce random sets, which we will callBernoullirandom sets.

Definition 3.1. • ABernoullirandom vectorϵ⁽ⁿ⁾of parameterp ∈(0,1)is a ran-dom vector whose entriesϵj,j ∈[n]are independent copies of aBernoulli ran-dom variable taking the values1,−1with probabilityp,(1−p), respectively.

• ABernoullirandom matrix E^(N,n) of parameterp ∈ (0,1) is anN ×n random matrix where each row is an independent copy of aBernoulli random vector ϵ⁽ⁿ⁾of parameterp.

• For any finite set J, theBernoullirandom set S^(J) of parameterp ∈ (0,1)is a random subset ofJ, such that for anyA⊂ J,

Pf

S^(J)=Ag

=p^|A|(1−p)^|J|−|A|. (3.1) That is, each element is included with probabilityp.

Remark 3.2. For all random variables described above, we will sometimes omit the upper indices if they are clear from the context. Furthermore, we writeqinstead of1−p andS⁽ⁿ⁾instead ofS^([n]).

The connection betweenBernoullirandom vectors and matrices andBernoulli ran-dom sets is evident from their definition.

Remark 3.3. Letϵ⁽ⁿ⁾ be aBernoullirandom vector with parameterp. Then the ran-dom set

S⁽ⁿ⁾ =(

j|ϵ_j =1)

⊂ [n]

is aBernoulli random set with the same parameter; we will callS⁽ⁿ⁾ theBernoulli random set corresponding toϵ⁽ⁿ⁾. Also note that, since for arbitrary finite set J and a Bernoullirandom setS^(J)of parameterpit holds that

X

it follows that (3.1) actually defines a probability distribution.

With the next lemma we can transfer the problem of bounding small ball probabilities as in(1.1)to the domain ofBernoullirandom sets and (symmetric)Sperner-kfamilies;

it generalizes the ideas used byErdősto prove the Lemma ofLittlewoodandOfford in [Erd45].

Lemma 3.4. Letϵ⁽ⁿ⁾be aBernoullirandom vector with parameterpand letS⁽ⁿ⁾be the correspondingBernoullirandom set. IfV is the union ofk open intervals of length at most2c andx ∈^Rnis an arbitrary vector withmin_j∈[n]|x_j| ≥c >0, then

IfV is symmetric, we only need to consider symmetricSperner-k families; we denote the corresponding probability byP_±,k,n(p).

Proof. We will only prove the theorem in the case whereV is symmetric; the general

Lemma 2.12 now implies thatB^sgn(x) is a subfamily of a symmetricSperner-k family.

Since by construction, it holds that

he,xi ∈V ⇔ eis a row ofE ⇔ {j|e_j =1} ∈^B, it follows with Remark 3.3 that

Pf

hϵ⁽ⁿ⁾,xi ∈Vg

=^Pf

S⁽ⁿ⁾ ∈^Bg

. (3.2)

Bearing in mind that the familyB^sgn(x)is a subfamily of a symmetricSperner-kfamily and that

This completes the proof.

Remark 3.5. As in Remark 2.17, Lemma 3.4 implies for aBernoullirandom vectorϵ⁽ⁿ⁾ of parameterp, arbitrarys-sparsex ∈ ^Rn, and arbitrary discrete setV ⊂ ^R, |V| = k

whereJ =supp(x). IfV is symmetric, we similarly obtain Pf

The quantitiesP_k,n(p)andP_±,k,n(p) will play an important role in the remainder of the chapter. A key distinction between the general and the symmetric case is that for the latter case, the following basic montonicity property is no longer true in general, see Remark 3.11 below.

Lemma 3.6. For arbitrary integerskandm ≤n and arbitraryp ∈(0,1), it holds that P_k,m(p) ≥ P_k,n(p).

Proof. By induction, it is enough to prove the statement form = n−1. For arbitrary A⊂ 2ⁿ, define

A₀={A⊂[n−1]|A∈^A} ⊂2ⁿ⁻¹ and A₁ ={A⊂ [n−1]|A∪ {n} ∈^A} ⊂2ⁿ⁻¹. IfAis aSperner-kfamily, then bothA₀ ⊂ 2ⁿ⁻¹andA₁ ⊂2ⁿ⁻¹areSperner-kfamilies.

Now letS⁽ⁿ⁾andS⁽ⁿ⁻¹⁾beBernoullirandom sets with parameterp,ξ ∈ {±1}ⁿbe a sign pattern andν ∈ {±1}ⁿ⁻¹the restriction ofξ to the firstn−1entries. Ifξ_n =1, we have

Pf

S⁽ⁿ⁾ ∈^Aξg

=q·^Pf

S⁽ⁿ⁻¹⁾ ∈^A0^ν

g

+p·^Pf

Sⁿ⁻¹ ∈^A1^ν

g ≤pPk,n−1 +qPk,n−1 =Pk,n−1. (3.3) Ifξ_n =−1, inequality(3.3)holds true with interchanged roles ofpandq. This completes

the proof.

The next definition is required in order to be able to transfer the ideas of Corollary 2.16 to the random matrix case.

Definition 3.7. LetF_k,n ⊂ 2²ⁿ be the set of allmaximal modulated Sperner-kfamilies, that is, the set of all modulatedSperner-k familiesA ⊂ 2ⁿ which are not a proper subfamily of any other modulatedSperner-k family.

Furthermore, denote by F_±,k,n ⊂ 2²ⁿ the set of all maximal modulated symmetric Sperner-kfamilies.

Remark 3.8. Definition 3.7 also enables us to rewrite the probabilityP_k,n(p) in terms of the setF_k,n, since for aBernoullirandom setS⁽ⁿ⁾with parameterpit holds that

P_k,n(p) = max

ξ∈ {±1}ⁿ max

A⊂2ⁿ ASperner-k

Pf

S⁽ⁿ⁾∈^Aξg

= max

A∈F_k,n

Pf

S⁽ⁿ⁾ ∈^Ag ,

and similarly forP_±,k,n(p)andF_±,k,n.

For arbitraryp ∈ (0,1) we will now compute the probabilitiesP_1,2(p) andP_±,2,2(p), which we will need later.

Lemma 3.9. For arbitraryp ∈ (0,1), it holds that

P_±,2,2(p)=P_1,2(p)=p²+q².

Proof. Letp ∈ (0,1) be arbitrary and letFbe the set of allSperner familiesA ⊂ 2². Then

F= (

∅,{∅},{{1}},{{2}},{{1,2}},{{1},{2}}) .

Direct calculations yield that the set of all modulatedSperner families of subsets of {1,2}is given by

F^′ =F∪(

{∅,{1,2}}) .

Consequently, the set of all maximal modulatedSpernerfamilies is given by F_1,2= (

{{1},{2}},{∅,{1,2}}) .

Next, we will computeF_±,2,2. By Definition 2.10, Definition 2.14 and Remark 2.6, every modulated symmetricSperner-2family of subsets of[n]is of the form

A∪^A−1ξ

=^Aξ ∪^A−ξ,

whereA ⊂2ⁿis aSperner-1family andξ ∈ {±1}ⁿ is a sign pattern. Since forA∈F_1,2, one hasA=^A−1and hence

F_±,2,2= (

A∪^A−1A ∈F_1,2)

=F_1,2,

we can conclude thatP_±,2,2(p) =P1,2(p). It remains to show thatP1,2(p) =p²+q². For that, note that

P_1,2(p) = max

A∈F1,2

Pf

S⁽²⁾∈^Ag

=max(

2pq,p²+q²)

=p²+q²,

where the last equality is implied by

p²+q²−2pq =(p−q)² ≥0.

This completes the proof.

Lemma 3.10. We have

|F_±,2,3| =4, and for arbitraryp ∈(0,1), it holds that

P_±,2,3(p)=1−pq.

Remark 3.11. While the probabilitiesP_k,n(p)are non-increasing with respect ton(Lemma 3.6), the same does not necessarily hold true forP_±,k,n(p). Lemma 3.9 and Lemma 3.10 imply for arbitraryp ∈(0,1)that

P_±,2,3(p)−P_±,2,2(p) =1−pq−(p²+q²)= (p+q)²−pq−(p²+q²)=pq >0.

Proof of Lemma 3.10. Since the families A₁= (

{1},{2},{3})

and A₁⁻¹ = (

{2,3},{1,3},{1,2}) areSpernerfamilies, the set

F^′= (

A₁^ξ ∪^A1^−ξξ ∈ {±1}³)

consists of modulated symmetricSperner-2families. We claim thatF^′=F_±,2,3. To this end, observe that by union compatibility, it holds for arbitraryξ ∈ {±1}³that

A₁∪^A1⁻¹

ξ

=

2³\ {∅,{1,2,3}}ξ

= (2³)^ξ \ {∅,{1,2,3}}^ξ = (2³)\ {∅,{1,2,3}}^ξ. Since{∅^ξ|ξ ∈ {±1}ⁿ}=2ⁿ, it therefore follows that

F^′= (

2³\ {A,A^c}A∈2³ )

= (

2³\ {A,A^c}A∈2³, |A| ≤ 1 )

, (3.4)

where the last equality holds by symmetry of the set{^A,A⁻¹}. Consequently, all the modulated symmetricSperner-2families contained inF^′are maximal. Indeed, for any A ∈ F^′, adding some A ∈ 2³ \^A results in a family which is not symmetric, and adding both missing sets results in2³, which is not a modulated symmetricSperner-2 family. The same arguments also show that there are no modulated symmetric Sperner-2families of cardinality larger than 6. For a modulated symmetric Sperner-2family A ∈ 2³ of cardinality smaller than6, its complement must also be symmetric, which shows thatAis a subfamily of someA^′ ∈F^′. Hence,A cannot be maximal and one hasF_±,2,3 =F^′. Since eachA ∈ 2³with|A| ≤ 1yields a different set2³\ {A,A^c},(3.4)

implies that|F_±,2,3| = |{A⊂ 2³||A| ≤ 1}| = 4. For the second part, a direct calculation

This completes the proof.

The next lemma transfers the ideas of Corollary 2.16 to the random matrix case. It will enable us to prove a more general version of Theorem 1.2 in terms of the quantities P_k,n(p)andP_±,k,n(p)for a constant number of columns.

Lemma 3.12. LetE^(N,n)be aBernoullirandom matrix with parameterp, let furtherV be the union ofk open intervals of length at most 2c, and letℓbe an integer. Then with probabilityPof at most

ℓ

is unique, then the probabilityP is bounded by

|F_k,s1| n

Proof. We again only establish the lemma for the symmetric case, as the general case is analogous. Note that each row of the random matrixE^(N,n)is an independent copy of aBernoullirandom vectorϵ⁽ⁿ⁾. The familyS⁽ⁿ⁾ ⊂2ⁿcontaining the sets

S_i⁽ⁿ⁾ = (

j^Ei,j(N,n) =1)

, i ∈[N],

therefore is a random family of independent Bernoulli random sets with the same parameterpasE^(N,n). Recall thatV is a union ofkopen intervals of length at most2c and symmetric. Now consider the families

(S⁽ⁿ⁾⊓JJ ⊂ [N],k ≤ |J| ≤ℓ)

. (3.8)

Applying a union bound over allJ ⊂ [n]withk ≤ |J| ≤ ℓ, we can therefore estimate the probabilityP thatE^(N,n)x ∈Vⁿfor somexwithk ≤ kxk0 ≤ ℓusing Corollary 2.16 as

P ≤^Pthere exists a modulated symmetricSperner-k family in (3.8)

≤ X

J⊂2^s k≤ |J| ≤ℓ

Pf(

S_i⁽ⁿ⁾⊓Ji ∈[N])

is a modulated symmetricSperner-k familyg

≤

is a modulated symmetricSperner-kfamilyg

≤

The last inequality holds since every modulated symmetricSperner-k family is a sub-family of a maximal modulated symmetricSperner-kfamily. By another union bound, it follows that

which establishes the first part of the lemma. For the second part, note that assumption (3.7)implies that there existsQ <P_±,k,s1 such that

max

k≤s≤ℓ s,s1

P_±,k,s ≤Q.

Recall from Remark 3.8 that

where we used the well-known inequality n

The next lemma allows us to prove a generalized version of Theorem 1.2 for vectorsx withkxk0 ≥s₀. In contrast to Lemma 3.12, it only considers discrete setsV, but allows the number of columnsnto tend to infinity. The proof is based on ideas byOdlyzkoin [Odl88].

Lemma 3.13. LetM ∼ ^E(N,n) be aBernoullirandom matrix with parameterp ∈ (0,1) and assume there exists a constantδ ∈(0,1)with

µ :=min{p,q} ≥N^−(1−δ). (3.11) Furthermore, assume that for constant integersk,s0there exists aQwith

Q ≥

Then there exists an absolute constantC >0, depending only onkandδ, such that for any setV ofkdistinct points, for

n ≤ 1− C log(N)

!

N (3.14)

and for arbitraryQ¯ >Q, the probabilityPthat there exists a vectorx ∈^Rnwithkxk0 ≥s₀ such thatMx ∈V^N satisfies

P =o Q¯^N

. IfV is symmetric, we can replacePk,s withP_±,k,s in(3.13).

Remark 3.14. In the non-symmetric formulation of Lemma 3.13, we can replace as-sumption (3.13)with Q ≥ p

P_k,s0(p), as P_k,n(p) is monotone with respect ton, see Lemma 3.6. This is not possible in the symmetric formulation, since, as noted in Re-mark 3.11,P_±,k,n(p) is not monotone with respect ton.

Remark 3.15. Note that the number of columnsn in Lemma 3.13 is allowed to tend to∞, as long as(3.14)is satisfied. The lower bound on the minimum ofpandqis not an artifact of the proof, but it is necessary in order to ensure that the probabilityP in Lemma 3.13 tends to0asN,n → ∞. To see this, letE:=^E(N,n)be aBernoullirandom matrix of parameterp = _N^c for some constantc > 0independent ofN. Considering onlys₀fixed columns ofE, we have

Pf

E_i,j =−1∀i∈[N], j ∈[s₀]g

=q^s0N =

1− _N^c s0N

≥ ₂¹e^−cs0, (3.15) for N large enough. Suppose now that there exists an index ℓ ∈ [⌈ⁿ/s0⌉] such that E_i,j =−1for alli ∈[N]andj ∈ {s₀ℓ+1, . . . ,s₀(ℓ+1)}, which happens by independence and(3.15)with probability at least

1−

1− 2¹e^−cs0_⌈sⁿ

0⌉

. (3.16)

Then, for arbitraryy ∈^Rand thes₀-sparse vectorx ∈^Rndefined via xj =



−_s^y0 j ∈ {s₀ℓ+1, . . . ,s₀(ℓ+1)},

0 else,

we have (Ex)_i =y for alli ∈[N]. Therefore, the probabilityP in Lemma 3.13 must be larger than(3.16), which tends to1forn→ ∞.

Proof of Lemma 3.13. Again, we only establish the proof for symmetricV, the gen-eral case is analogous by replacing all occurring symmetric Sperner-k families with Sperner-k families. Throughout this proof, we will omit the argumentpofP_±,k,n. Let

M ∼^E(N,s)be a randomBernoullimatrix with parameterpand letQ_±,N,sbe the prob-ability that there exists a vectorx ∈^Rswith non-vanishing entries such thatMx ∈V^N. We can now apply a union bound over the possible supports, to estimate

P ≤

be the probability that the matrixM⁽¹⁾ ∼ ^E(s+ℓ,s) does not have full rank; we consider this case separately. Suppose now thatM⁽¹⁾is injective. Then there existsR ⊂ [s +ℓ]

with |R| = s, such that the restrictionM˜⁽¹⁾ ofM⁽¹⁾ to the rows indexed byR has full rank. For each of thek^s vectorsy ∈ V^s, there hence exists a unique vectorx ∈ ^Rs withM˜⁽¹⁾x = y. By invertibility of M˜⁽¹⁾, the case of x ∈ ^Rs with vanishing entries does not contribute toQ_±,N,s, and we can assume thatxonly has nonvanishing entries.

By stochastic independence of the rows, we can therefore bound the probability that M⁽²⁾x ∈ V^{(N−s−ℓ)} from above byP˜^N−(s+ℓ)

±,k,s . Here,P˜_±,k,s is the probability that for a Bernoullirandom vectorϵ⁽ⁿ⁾of parameterp, it holds thathe⁽ⁿ⁾,xi ∈V. By Remark 3.5, P˜_±,k,s can be estimated as

P˜_±,k,s ≤P_±,k,s ≤Q²; (3.18)

the last inequality holds by assumption(3.13). On the event thatM⁽¹⁾ is injective, we apply a union bound over all _s+ℓ

s

candidates for a subsetRand allk^s vectorsy ∈V^N. Combining this with the event thatM⁽¹⁾is not injective we can therefore bound

Q_±,N,s ≤ s+ℓ s

!

k^sP˜^N−(s+ℓ)

±,k,s +Q_s+ℓ,s. (3.19)

We will now boundQs+ℓ,s using a union bound over all potential ranks ofM⁽¹⁾ and an approach similar to the one above. Suppose thatM⁽¹⁾ has rankr with1 ≤ r ≤ s −1, matrixM_:,C⁽¹⁾. The vectorx cannot have any vanishing entries, since this would imply thatM⁽¹⁾had rank smaller thanr. AsM_:,C⁽¹⁾ has rankr, there existsR ⊂ [s+ℓ],|R| =r, such thatM_R,C⁽¹⁾ is invertible; the vectorx in(3.20)is therefore unique. Note that(3.20) also implies

M_R⁽¹⁾c,C∪{τ}(x^T,−1)^T =0. (3.21)

Consequently,

the last line follows by a union bound over all choices ofC,Rwhile always choosing the smallestτin[s]\C. Conditioning onM_R,C⁽¹⁾ , which is invertible,rank(M_:,C⁽¹⁾

∪{τ}) =r can only hold true if the uniquex in(3.20)also satisfies(3.21). By stochastic independence of the rows ofM_R⁽¹⁾c,C, we therefore have

whereP˜1,r+1is the maximal probability that forxas in(3.21), and aBernoullirandom vectore⁽ⁿ⁾ of the parameterp, it holds thathe⁽ⁿ⁾,(x,−1)^Ti = 0. By Remark 3.5,P˜1,r+1

can be estimated as

P˜1,r+1 ≤ P1,r+1 ≤P1,2=p²+q² ≤Q²; (3.24) the inequalities hold by Lemma 3.6, Lemma 3.9, and(3.12). Bringing(3.22) and(3.23) together, we therefore arrive at

Qs+ℓ,s ≤

Applying(3.19)together with(3.18),(3.26), andn,s+ℓ≤ N, it follows that

In last inequality, we used(3.12)to bound

Q^−2Nε ≤ (p²+q²)^−Nε =(1−2pq)^−Nε ≤ (¹/2)^−Nε =2^Nε. (3.30) Using monotonicity and inequality(3.30), we can now bound

log(S₁)−log(Q¯^N)≤ log(D_N)−log(Q¯^N) (3.31) M⁽²⁾, consisting of the firstn+ℓrows ofM, and the remaining rows, respectively, for some ℓ ≤ N −n to be determined later. If M⁽¹⁾ is injective, so are all of its column restricted submatrices. Hence, on the event thatM⁽¹⁾ is injective (the complementing event has probability at mostQ_n+ℓ,n), we obtain a bound similar to(3.19)but without Q_s+ℓ,s. It follows that

Using Lemma 1.5 and a union bound over thekpoints inV, we can now bound P˜_±,k,s ≤P˜_k,s ≤ kC

õs. (3.33)

It follows from(3.32)that

Applying again(3.33), we find that T₂≤n²4^N C

where we used(3.10)in the second and(3.30)in the third inequality. If we chooseℓsuch thatn+ℓ≥ ^N/2, we obtain, withD_N as in(3.29)and(3.31), that

Ifn ≤ ^N/2, we can chooseℓ = max{^N/2−n,^N/4}, which implies via direct calculation

Suppose for a moment that

kC

Then, from(3.35), it follows that S2≤o

To complete the proof, note that by the monotonicity of the logarithm,(3.36)is equival-ent to

ℓ > log(9√

2)N

δ/4log(N)−log(kC),

which, forN > (kC)^4/δ and a sufficiently large constantC˜ > 0depending only on the constantsδ andk, is implied by

ℓ ≥ CN˜

In this section, we reduced the problem of investigating the span of the columns of a randomBernoullimatrixE^(N,n)to the problem of bounding the quantitiesP_k,n(p)and P_±,k,n(p), which we aim to bound in the remainder of the chapter. In Section 4, we will boundP_±,k,n(p) andP_±,k,n(p)using cardinality estimates forSperner-k families and a greedy method. Since these bounds are only applicable for certain values ofpandn, we

will reduce the problem of boundingP_k,n(p)andP_±,k,n(p)in Section 5 in a way that we can apply the LYM-inequality, which we will also introduce in that section. Whereas resulting bounds onP_±,k,n(p)will be weaker than the bounds in Section 4, they will be monotone with respect ton.

4. Bounding P P P

_kkk,,,nnn

and P P P

_±±±,,,kkk,,,nnn

using Cardinality Estimates

In the case wherek = 1, the problem of bounding the cardinality of aSpernerfamily was first addressed bySperner.

Lemma 4.1 (Sperner’s Lemma [Spe28]). LetA⊂ 2ⁿ be aSpernerfamily, then

|^A| ≤ n

⌊ⁿ/2⌋

! .

In particular, ifnis even, the largestSpernerfamily of[n]contains exactly the subsets of cardinalityⁿ/2of[n]. Ifn is odd, the largest two Spernerfamilies of[n]are the families containing all subsets of cardinality⌊ⁿ/2⌋of[n]or all subsets of cardinality⌈ⁿ/2⌉of[n].

Sperner’s Lemma can be generalized toSperner-kfamilies. The corresponding result is considered folklore without known reference, see, e.g., [EFK05].

Lemma 4.2. LetA⊂ 2ⁿ be aSperner-kfamily. Then

|^A| ≤

⌊(n+k−1)/2⌋

X

i=⌊(n−k+1)/2⌋

n i

! .

Equality holds if and only ifAis the family of all setsAwith |A| ∈[⌊^(n−k2⁺¹⁾⌋,⌊^(n+k2⁻¹⁾⌋] or the family of all setsAwithA∈[⌈⁽ⁿ⁻2^k+1)⌉,⌈^(n+k2⁻¹⁾⌉].

In order to be able to handle symmetricSperner-2families, a refinement ofSperner’s Lemma will be useful. The following lemma, which is due toMilner, gives an estimate for the cardinality of a special class ofSpernerfamilies.

Lemma 4.3 (Milner[Mil68]). LetA ⊂2ⁿbe aSpernerfamily which does not contain any complementing sets, i.e., setsA∈^AwithA^c ∈^A. Then

|^A| ≤ h_nn₋₁

2

i! .

We can now use Lemma 4.3 to estimate the cardinality of symmetricSperner-2 fam-ilies.

Corollary 4.4. LetA⊂ 2ⁿ be a symmetricSperner-2family. Then

|^A| ≤ 2 n

⌊⁽ⁿ⁻¹⁾/2⌋

! .

Proof. LetA ⊂2ⁿbe an arbitrary symmetricSperner-2family. By Definition 2.10,A is of the form

A=^B∪^B−1,

whereB ⊂ 2ⁿ is aSpernerfamily. We may assume thatBdoes not contain any com-plementing sets. Lemma 4.3 now implies that

|^A| ≤2|^B| ≤2 n

⌊⁽ⁿ⁻¹⁾/2⌋

! .

This completes the proof.

ForBernoullirandom sets of parameterp =¹/2, we have the following:

Lemma 4.5. LetA ⊂ 2ⁿ be a family andS⁽ⁿ⁾ aBernoullirandom set with parameter p =¹/2. Then it holds for arbitrary sign patternξ ∈ {±1}ⁿthat

Pf

S⁽ⁿ⁾ ∈^Aξg

= |^A| 2ⁿ .

Proof. ABernoullirandom setS⁽ⁿ⁾of parameterp = ¹/2attains each setA∈2ⁿwith the same probability2⁻ⁿ. It follows that

Pf Together with the cardinality bounds above, it is now straightforward to estimate the probabilitiesP_±,k,n(p)andP_k,n(p)in the case ofp =¹/2. Note that Theorem 1.3 was proven byErdősin [Erd45] usingSperner’s Lemma (Lemma 4.1) and the observation of Lemma 4.5. While this will eventually give rise to generalized Littlewood-Offord-type inequalities using Lemma 3.4, it also directly yields Theorem 1.2 forp= ¹/2, which basically is the case considered byOdlyzko(Theorem 1.1).

Proof of Theorem 1.2 forp =¹/². Sincep = ¹/2is fixed, we will omit the argumentp forP_±,k,n and P_k,n. Note that {±1} is a symmetric set of two points. Lemma 3.9 and Lemma 3.10 now yield

P_±,2,2=¹/2, P_±,2,3=³/4.

For alls ≥4, Lemma 3.6 and Lemma 4.5 together with Lemma 4.2 imply

P_±,2,s ≤P2,s ≤P2,4≤ Similarly, it holds for alls ≥6that

P_±,2,s ≤ P2,6≤

Each of these bounds will now be used to bound one part of the probability. Denote byP1 the probability that there exists a vectorx ∈ ^Rn with2 ≤ kxk⁰ ≤ 5such that

and the maximizer is unique. Observing that|F_±,2,3|=4(Lemma 3.10), Lemma 3.12 now implies that

In the proof of Theorem 1.2, we did not have to fully exploit the symmetry of the set {±1}, since we used the upper boundP_±,2,n(¹/2) ≤P_2,n(¹/2)forn ≥ 4. In order to be able to prove Theorem 1.2 for arbitraryp∈ (0,1), the symmetry of the set{±1}will be crucial.

Lemma 4.6. Let S⁽ⁿ⁾ be a Bernoulli random set with parameter p , ¹/2 and let A₁,A₂ ⊂[n]be arbitrary subsets of cardinalityk₁,k₂resp. withk₁,k₂ ≤ⁿ/2. Then

Pf

S⁽ⁿ⁾ ∈ {A₁,A^c₁}g

≥^Pf

S⁽ⁿ⁾ ∈ {A₂,A^c₂}g if and only if

k₁ ≤k₂.

Proof. By the definition of theBernoullirandom setS⁽ⁿ⁾, it follows forj =1,2that Pf

S ∈ {A_j,A^c_j}g

=p^kjq^n−kj +p^n−kjq^kj; (4.2)

Without loss of generality, we may assume thatp >q. Now consider the difference of the respective probabilities,

P

S ∈ {A₁,A^c₁}

−^P

S ∈ {A₂,A^c₂}

=p^k1q^n−k1 +q^k1p^n−k1 −p^k2q^n−k2 −q^k2p^n−k2

=

p^k2q^n−k2 1− _p

q

n−k1−k2 _p

q

k1−k2

−1

.

(4.3)

The first factor on the right hand side of(4.3)is strictly positive. Ifk1 ≥k2, then both the second and the third factor in(4.3)are non-positive. Ifk₂ <k₁, the second factor is negative (this follows fromk1 ≤ ⁿ/2) and the third factor is positive. Consequently, the left hand side of(4.3)is non-negative if and only ifk1 ≤k2. This completes the proof.

To translate this into an upper bound onP_±,2,n(p), we need the following lemma.

Lemma 4.7. Letn ≥2,A ⊂2ⁿbe a symmetricSperner-2family and denote byL_j ⊂ 2ⁿ, j ∈[n], the family of all setsA∈2ⁿ with|A|=j. Then

N:=^L0∪^L1∪^Ln−1∪^Ln

is not a subfamily ofA^ξ for any sign patternξ ∈ {±1}ⁿ.

Proof. Suppose thatN is a subfamily ofA^ξ for an arbitraryξ ∈ {±1}ⁿ, which, since Ais a symmetricSperner-2family, implies that there exists a subfamilyB⊂ ^Aξ with B∩^B−1=∅such that

N =^L0∪^L1∪^Ln−1∪^Ln =^B∪^B−1. (4.4) Using the identity(A^ξ)^ξ = ^A, we can conclude thatB^ξ and thusB^−ξ areSperner families. We will now consider the two occurring cases in more detail. In the first case, BorB⁻¹contains∅and at least one set of cardinality1. If∅ ∈^Band no set of cardinality 1is contained inB, it follows thatL₁ ⊂ ^B−1, which also implies thatL₁⁻¹ = ^Ln−1 ⊂ ^B. By interchanging the roles ofBand B⁻¹, it follows that the remaining second case is the one where eitherB orB⁻¹is equal toL₀∪^Ln−1. By symmetry, it is in both cases enough to only consider the familyB. In the first case, suppose that there exists an indexi ∈ [n]such that {∅,{i}} ⊂ ^B. For the symmetric difference of the two sets, Proposition 2.8 implies that

∅^ξ∆{i}^ξ =∅∆{i} ={i},

i.e., it either must hold that∅^ξ ={i}^ξ ∪· {i}or that{i}^ξ =∅^ξ ∪· {i}. Any of the two cases contradicts the fact thatB^ξ is aSpernerfamily, since{i}^ξ ⊂ ∅^ξ or∅^ξ ⊂ {i}^ξ. We can analogously handle the case whereB⁻¹contains∅and one subset of cardinality1. Next, suppose thatB =^L0∪^Ln−1. Again by Proposition 2.8, it holds that for any setB ∈^Ln−1

|∅^ξ∆B^ξ| = |∅∆B|= |B|=n−1.

Consequently,B^ξ must contain|^Ln−1|=ndistinct setsBwith|∅^ξ∆B| =n−1, as(·)^ξ is a bijection of[n]onto itself. We claim that this already contradicts the assumption that B^ξ is aSpernerfamily. If∅^ξ is equal to∅or[n],B^ξ with|^Bξ| ≥2cannot be aSperner family. So suppose that|∅^ξ| = kwith1 ≤ k ≤ n−1. Note that|∅^ξ∆B^ξ| =n −1either implies thatB^ξ and∅^ξ are disjoint with|B^ξ|=n−k−1, or thatB^ξ is intersecting∅^ξ in a single element and it holds that|B^ξ| =n−k+1. In2ⁿ, there existn−ksetsB^ξ for which the first assumption is satisfied andksetsB^ξ for which the second one holds. SinceB^ξ is aSpernerfamily, it cannot contain both a setB₁of the first type and a setB₂of the second type, as this would imply thatB₁ ( (∅^ξ)^c ( B₂. Therefore, it contains at most max{k,n−k}subsetsB^ξ with|∅^ξ∆B^ξ| =n−1. This yields the desired contradiction and

completes the proof.

We can now finally derive a strong upper bound on P_±,2,3(p) for allp ∈ (0,1) and smallnusing a greedy approach.

Lemma 4.8. LetS⁽ⁿ⁾be a randomBernoulliset with parameterpand let furtherA ⊂2ⁿ be a symmetricSperner-2family and letξ ∈ {±1}ⁿan arbitrary sign pattern. Then

Pf

S ∈^Aξg

≤^P[S ∈^B], (4.5)

whereBis the subfamily of2ⁿ\ {{1},{1}^c}consisting of the _n

⌊⁽ⁿ⁻¹⁾/²⌋

sets that are smallest and largest in cardinality.

In particular, forn ∈ {2,4,5,6}andp∈ (0,1), it holds that P_±,2,n(p) <P_±,2,3(p).

Proof. LetA⊂2ⁿbe a symmetricSperner-2family. By Corollary 4.4, it follows that

|^Aξ| ≤ 2 n

⌊⁽ⁿ⁻¹⁾/2⌋

! .

Ifp=¹/2, the assertion of the lemma follows from Lemma 4.5; we may therefore assume thatp, ¹/2. In addition to the cardinality constraint, Lemma 4.7 implies that, ifn ≥ 2, the family

N=^L0∪^L1∪^Ln−1∪^Ln

is not a subfamily of A^ξ. Furthermore, since A is a symmetricSperner-2family, it holds that A = ^A−1. Using these three observations, we can now obtain the upper bound holds if and only if

k1 ≤k2.

If we neglect the constraintN1Bon the right hand side of(4.6)and only consider the cardinality and symmetry constraints, we can therefore construct a maximizer Cin a greedy manner by selecting the _n

⌊⁽ⁿ⁻¹⁾/²⌋

sets of smallest cardinality and their comple-ments. However, forn ≥5, a family constructed in this way will always be a superfamily ofNand will thus violate the subfamily constraint. Again by Lemma 4.6, the family of

sets of smallest cardinality and their comple-ments. However, forn ≥5, a family constructed in this way will always be a superfamily ofNand will thus violate the subfamily constraint. Again by Lemma 4.6, the family of

Im Dokument On two Random Models in Data Analysis (Seite 35-0)