Restriction to manageable sets

(a) G_nⁱ is a grid,

(b) Rⁱ_n is a refinement of the grid G_nⁱ.

(c) S_nⁱ ⊂ Rⁱ_n satisfies G1, G2 ∈ G_nⁱ =⇒ 0 <

S(S_nⁱ)^G1 =

S(S_nⁱ)^G2 <

|G₁|=|G₂|.

Notation 11.3. In this situation we denote by cⁱ_n ∈(0,1) the coverage ratio ofS_nⁱ, a number satisfying G∈ G_nⁱ =⇒

S(S_nⁱ)^G

=cⁱ_n|G|. We also denote lⁱ_n= 1−cⁱ_n and call this number theloss ratio of S_nⁱ. Furthermore we denote

S_nⁱ := [ S_nⁱ S_n :=

n

\

k=1 N(k)

[

i=1

S_kⁱ

S_n,i := Sn−1∩ S_n¹ ∪...∪S_nⁱ

(whereS₀:=[0,1]^d ) S_n,i :=

R ∈ Rⁱ_n|R ⊂S_n,i T (N,G,R,S) :=

∞

\

n=1

S_n=

∞

\

n=1 N(n)

[

i=1

S_nⁱ =

∞

\

n=1 N(n)

[

i=1

S_n,i.

Definition 11.4. We say that S ⊂[0,1]^d is a set of type N when there exists a scheme (N,G,R,S) of type N such thatS ⊂T (N,G,R,S).

Notation. As in the case of H^(N)-sets, we will mostly be interested in the sets S which satisfy S = T (N,G,R,S). When a scheme of type N uses a different letters, i.e. (N,C,B,A), we will obviously not denote the respective sets by G_nⁱ, S_nⁱ, S etc. but rather by C_nⁱ, Aⁱ_n, A etc. In order to avoid confusion, we will sometimes add the name of the set as an index to the related variables, i.e.

l_nⁱ =lⁱ_n,A for the loss ratio of C_nⁱ.

Figure 7: First two steps of a construction of a type 1 setS: S₁¹forG₁¹ =

[0,1]² , R¹₁ =

0 +i,¹₂ +i

×

0 +j,¹₂ +j

|i, j ∈ 0,¹₂

and c¹₁ = ¹₂ (left). S₂¹ with c¹₂ = ⁴₉ using the grid and refinement from Figure 6 (right). For an example of a

”nicer” scheme see Figure 8.

G be a grid and let G∈ G be it’s element. If a system Mof measurable non-flat subsets of R^d satisfies ∀M ∈ M : diamM ≥ CdrkGk, then we have ∀M ∈ M :

#G^M ≥cmr|M|/|G|.

Proof. LetG∈ G, M ∈ Mbe sets and x∈M the point from the definition of a non-flat set. By symbolB_η =B(x, η) we will denote the closed balls centered at this point. We haveB_r ⊂M ⊂B_R for some r, R >0 satisfying _R^r ≥N⁻²

f . Clearly we have

G∩B_r−diamG6=∅ =⇒ G⊂M. (10) Thus we can bound the number #G^M:

#G^M =

SG^M

|G| ≥

SG^Br

|G|

(10)

≥

B_r−kGk

|G| = |M|

|G|

B_r−kGk

|B_r|

|Br|

|M|

≥ |M|

|G|

Br−kGk

|B_r|

|B_r|

|B_R| = |M|

|G|

r− kGk r

d

r R

d

.

We have ^r−kGk_r = 1− ^kGk diamM

diamM

r ≥1− ^kGk

diamM 2R

r = 1− ^kGk diamM2N²

f, and so for diam_kGk^M ≥= 4N²

f we get

#G^M ≥ |M|

|G|

1 2

d

N⁻²

f d

.

Therefore the numbers Cdr = 4Nf² and cmr = ¹₂d N⁻²

f d

are the desired

constants.

Notation 11.7. If two systems M and P of subsets of R^d (not necessarily grids) satisfy the inequality ∀M ∈ M : diamM ≥ CdrkPk from the previous lemma, we write M P. If they satisfy ∀M ∈ MdiamM > kPk (resp. ≥), we write M>P (resp. ≥). Clearly for any scheme (N,G,R,S) of typeN and anyn ∈N, i≤N(n) we have G_nⁱ >Rⁱ_n ≥ S_nⁱ.

Figure 8: First two steps of the construction of a regular set of type 1. S₁¹ (left) is the same as on Figure 7. S₂¹ (right) satisfies G₂¹ =

0 +i,¹₄ +i

×

0 +j,¹₄ +j

|i, j ∈₀

4, ...,³₄

, R¹₂ = 0 +i,¹₈ +i

×

0 +j,¹₈ +j

|i, j ∈₀

8, ...,⁷₈

and c¹₂ = ¹₄. Notice that G₂¹ re-fines R¹₁ and R¹₁ actually a grid, not just a refinement of G₁¹.

Definition 11.8. LetS ⊂T(N,G,R,S) be a set of type N ∈N∪ {∞}. We say that S is an L-set if it has the following key property

ˆ for each n ∈ N, i ≤ N(n) the systems G_nⁱ and Rⁱ_n consist of non-flat sets and we have

i < N(n) =⇒ Rⁱ_n\ S_nⁱ G_nⁱ⁺¹ and if the following technical conditions holds

1. Monotonicity of N: If N ∈ N, then N is constant. If N = ∞, then N is non-decreasing.

2. Measure loss control: For each i ∈N there exists lⁱ_S > 0 such that l_nⁱ ≥l_Sⁱ holds for each n ∈Nwith N(n)≥i.

3. The refinements are not too fine: For each i∈ N there exists dⁱ_S > 0 such that diamR ≥ dⁱ_SkG_nⁱk holds for each n ∈ N with N(n) ≥ i and each R∈ Rⁱ_n.

We say that S is a regular set if it has the following key property

ˆ for each n ∈ N, i ≤ N(n), Rⁱ_n is a grid and for each n, m∈ N, i ≤ N(n), j ≤N(m) we have⁹

(n, i)<(m, j) =⇒ G_m^j is a refinement of Rⁱ_n and if it satisfies the following technical conditions:

1. S is an L-set.

2. kRⁱ_nk decreases quickly: For any (n, i)>(1,1) we denote byδ_nⁱ the number satisfying kRⁱ⁻¹_n k = δ_nⁱ kRⁱ_nk (resp.

R^N(n)_n−1

= δ_n¹kR¹_nk for i = 1). Then the following two conditions hold:

(a) δ_n^{i n→∞}−→ ∞ holds for each i.

(b) For each n, m∈N, i≤N(n),j ≤N(m) we have (n, i)<(m, j) =⇒ Rⁱ_n G_m^j . 3. S is a ”true” set of type N: S =T (N,G,R,S).

Example 11.9. Let N ∈ N∪ {∞} and E ∈ H^(N). By Proposition 10.7 (1) we know that there exists some tuple (N,I,x) as in Notation 10.4, such that E ⊂ H(N,I,x), limN_n = N and x is quasi-independent. We can assume without loss of generality thatI_i = (a_i, b_i), where 0< a_i < b_i <1.

1. Every H^(N)-set is a set of type N:

We denote N = (N)^∞_n=1 and G = (G_nⁱ), R= (Rⁱ_n),S = (S_nⁱ), where G_nⁱ =

j/xⁱ_n,(j+ 1)/xⁱ_n

|0≤j ≤xⁱ_n−1 ,

S_nⁱ =

j/xⁱ_n,(j+a_i)/xⁱ_n

|j = 0, ..., xⁱ_n−1 ∪

∪

(j +b_i)/xⁱ_n,(j + 1)/xⁱ_n

|0≤j ≤xⁱ_n−1 , Rⁱ_n=S_nⁱ ∪

(j+a_i)/xⁱ_n,(j+b_i)/xⁱ_n

|0≤j ≤xⁱ_n−1 . These systems then witness thatE ⊂T (N,G,R,S) is a set of type N.

2. H_L^(N)

0 -sets are L-sets for L₀ ≥Cdr:

Suppose that the tuple (N,I,x) also witnesses thatH_L^(N₀⁾ for someL₀ ≥Cdr.

We will show that E is an L-set of type N. Clearly all of the systems from (N,G,R,S) consist of 1-dimensional intervals, i.e. of non-flat sets (in this case we could actually useNf = 1). By definition ofH_L^(N)

0 -set we havexⁱ⁺¹_n |I_i|/xⁱ_n ≥L₀

9Recall here the definition of ordering from Notation 10.8.

for i < N(n). Since kG_nⁱ⁺¹k = 1/xⁱ⁺¹_n and kRⁱ_n\ S_nⁱk = (b_i−a_i)/xⁱ_n = |I_i|/xⁱ_n, this means that for L₀ ≥Cdr we have

Rⁱ_n\ S_nⁱ /

G_nⁱ⁺¹

=xⁱ⁺¹_n |I_i|/xⁱ_n ≥L₀ ≥Cdr,

which implies that E satisfies the key property Rⁱ_n\ S_nⁱ G_nⁱ⁺¹. We also have l_nⁱ =lⁱ_E =b_i−a_i and dⁱ_A= min{a,_i b_i−a_i, 1−b_i}, thereforeE also satisfies the technical conditions necessary for being an L-set.

Existence of H^(N)-sets which are also regular sets of type N: Suppose now that E =H(N,I,x), E ∈ H_L^(N)₀ and for each i ≤ N we have Ii =

ai

qi,^b_qⁱ

i

, where 0 < a_j < b_j < q_i are integers, and assume that for each n ∈ N, i ≤N we have xⁱ_n ∈N and q_ixⁱ_n|xⁱ⁺¹_n (resp. q_Nx^N_n|x¹_n+1 in case that i=N).

By the previous point, E is an L-set of type N as witnessed by the scheme (N,G,R,S) from the first point. We will either show thatE is also a regular set or modifyE slightly so that it becomes a regular set (but remains a H^(N)-set).

Generally, the regularity of E cannot be witnessed by (N,G,R,S), since the systems Rⁱ_n are not necessarily grids. To remedy this we define ˜R:=

R˜ⁱ_n and S˜=

S˜_nⁱ

, where

R˜ⁱ_n :=

[j, j+ 1]/q_ixⁱ_n|j = 0, ..., q_ixⁱ_n−1 , S˜_nⁱ :=n

R∈R˜ⁱ_n|R ⊂[ S_nⁱo

.

The key property of being a regular set is then satisfied for these systems, since q_ixⁱ_n|x^j_m holds for (n, i)<(m, j). It remains to prove thatE satisfies the condition 2.

We have kG_nⁱk = 1/xⁱ_n and

R˜ⁱ_n =

S˜_nⁱ

= 1/qⁱ_nxⁱ_n and we know that xⁱ_n →

∞. We can assume that for everyi≤N we have xⁱ⁺¹_n /xⁱ_n→ ∞as n→ ∞ (resp.

x¹_n+1/x^N_n in case that i = N) - if this was not true, we could take a different set ˜E = H(N, I,x), where ˜˜ xⁱ⁺¹_n := xⁱ⁺¹_n xⁱ_n, ˜x¹_n+1 :=x¹_n+1x^N(n)n . This implies the technical condition 2.(a) which in turn implies that 2.(b) is satisfied for alln≥n₀ for some n₀ ∈N. Consequently the setE⁰ =H(N⁰,I,x⁰) where N⁰ = (N_n+n0)_n, x⁰ = (x_n+n0)_nis a H^(N)-set which is also a regular set of type N.

Note that by Corollary 10.11 we have

|E|= 0 ⇐⇒

E˜

= 0 ⇐⇒ |E⁰|= 0.

Remark 11.10 (H^(N)∗-sets). When E = H(N, I,x) ∈ H^(N)∗ \ H^(N), we have xⁱ_n ∈ R\ N for some n and i. For E ∈ H_L^(N)∗

0 , we can assume without loss of generality that xⁱ_n >2. Let N, G_nⁱ, S_nⁱ and Rⁱ_n be as in the representation of

H^(N)-sets. The tuple (N,G,R,S) is not a scheme of type N, because the systems G_nⁱ, Rⁱ_n only cover the interval [0,bxⁱ_nc/xⁱ_n] rather than the whole interval [0,1]

and we haveE (T

Sn. We set

G˜_nⁱ =G_nⁱ ∪ xⁱ_n

/xⁱ_n,1 , R˜ⁱ_n =Rⁱ_n∪

xⁱ_n

/xⁱ_n,1 , S˜_nⁱ =S_nⁱ ∪

xⁱ_n

/xⁱ_n,1 . This new tuple

N,G,˜ R,˜ S˜

is still not a scheme of type N since the elements of G_nⁱ necessarily do not have the same diameter, but we at least haveE∪ {1}= T

N,G,˜ R,˜ S˜ .

1. In order to avoid complicating the notation even further, we will not attempt to generalize the definition of sets of type N in such a way that H_L^(N)∗

0

-sets become L--sets of type N. However by using the representation ofH^(N)∗-sets introduced by this remark, we can still prove the main result (i.e. the existence of measureµ, supported onH^(∞)-set, which annihilates every H_C^(N)∗

dr-set) even for H_C^(N)∗

dr-sets. We claim that this will require only minor modifications to the proofs, namely the only affected proof is that of Proposition 11.19, where the addition of the interval [bxⁱ_nc/xⁱ_n,1] to the systems ˜S_nⁱ might slightly change the value of α. However since xⁱ_n >2, the new ˜α will not be lower than α/2^N. Consequently all of the later propositions will remain valid with the exact same proofs.

2. We only need theH^(N)∗-sets in order to be able to use the results from [Vla]

- but in fact, its author only uses rational quasi-independent sequences to prove his results. Therefore we can also observe that if a quasi-independent sequence x consists of rational numbers, it is possible to represent the resulting H^(N)∗-set as a set of type N. This gives us an alternative to the modification of our proves suggested in 1.

In document DIPLOMOV´A PR ´ACE (Stránka 51-56)