Slicing Sets and Measures, and the Dimension of Exceptional Parameters

Abstract

We consider the problem of slicing a compact metric space Ω with sets of the form $\pi_{\lambda}^{-1}\{t\}$, where the mappings π _λ :Ω→ℝ, λ∈ℝ, are generalized projections, introduced by Yuval Peres and Wilhelm Schlag in 2000. The basic question is: Assuming that Ω has Hausdorff dimension strictly greater than one, what is the dimension of the “typical” slice $\pi_{\lambda}^{-1}\{t\}$, as the parameters λ and t vary. In the special case of the mappings π _λ being orthogonal projections restricted to a compact set Ω⊂ℝ², the problem dates back to a 1954 paper by Marstrand; he proved that for almost every λ there exist positively many t∈ℝ such that $\dim\pi_{\lambda }^{-1}\{t\} = \dim\varOmega- 1$. For generalized projections, the same result was obtained 50 years later by Järvenpää, Järvenpää and Niemelä. In this paper, we improve the previously existing estimates by replacing the phrase “almost all λ” with a sharp bound for the dimension of the exceptional parameters.

Appendix A: Proof of Lemma 4.2

In this section, we provide the details for the proof of Lemma 4.2. The argument is the same as that used to prove [15, Lemma 4.6], and we claim no originality on this part. As mentioned right after Lemma 4.2, the only reason for reviewing the proof here is to make sure that the factor

produces no trouble—and in particular, no dependence on the index i∈ℕ!

Lemma A.1

Fix x,y∈Ω, x≠y, write r=d(x,y), and let I be a compact subinterval of J. The set

can be written as the countable union of disjoint (maximal) open intervals I ₁,I ₂,…⊂I.

• (i) The intervals I _j satisfy $\mathcal{L}^{1}(I_{j}) \leq2$. Furthermore, if I _j and I have no common boundary, then $r^{2\tau } \lesssim_{I,\tau} \mathcal{L}^{1}(I_{j})$. Thus, in fact, there are only finitely many intervals I _j .

• (ii) There exist points $\lambda_{j} \in\bar{I}_{j}$, which satisfy: if $\lambda\in\bar{I}_{j}$, then $|\varPhi_{\lambda}(x,y)| \geq|\varPhi_{\lambda_{j}}(x,y)|$ and |Φ _λ (x,y)|≥δ _I,τ r ^τ|λ−λ _j |. Furthermore, there exists a constant ε>0, depending only on I and τ, with the following properties: (a) if λ∈I _j and |Φ _λ (x,y)|≤δ _I,τ r ^τ/2, then (λ−εr ^2τ,λ+εr ^2τ)∩I⊂I _j , and (b) if $|\varPhi_{\lambda_{j}}(x,y)| \geq\delta_{I,\tau}r^{\tau}/2$, then |Φ _λ (x,y)|≥δ _I,τ r ^τ/2 for all λ∈(λ _j −εr ^2τ,λ _j +εr ^2τ)∩I.

Proof

Let J be one of the intervals I _j ; see Figure 2. According to the transversality condition (2.1), we have

for all λ∈J, which means that the mapping λ↦Φ _λ (x,y) is strictly monotonic on J. The first inequality in (i) follows from (2.1) via the mean value theorem; if [a,b]⊂J and ξ∈(a,b) is the point specified by the mean value theorem, we have

For the second inequality in (i) we apply the regularity condition (2.2); write J=(a,b). Since J and I have no common boundary, we have {Φ _a (x,y),Φ _b (x,y)}={−δ _I,τ r ^τ,δ _I,τ r ^τ}. Hence, by (2.2) and the mean value theorem,

Fig. 2

The interval J and some of the points λ _i

To prove (ii), let $\lambda_{j} \in\bar{I}_{j}$ be the unique point in $\bar{I}_{j}$ where the mapping λ↦|Φ _λ (x,y)| attains its minimum on I _j . Such a point exists by continuity and monotonicity; note that on all but possibly two of the intervals I _j (the left- and rightmost ones) λ _j is the unique zero of the mapping λ↦Φ _λ (x,y) on I _j . Now if λ∈I _j is any point, we see that Φ _λ (x,y) has the same sign and absolute value at least as great as $\varPhi_{\lambda_{j}}(x,y)$. This gives

by (2.1) and the mean value theorem. All that is left now are (a) and (b) of (iii); set $\varepsilon:= \delta_{I,\tau }C_{I,1,\tau}^{-1}/2 > 0$. Assume first that λ∈I _j , |Φ _λ (x,y)|≤δ _I,τ r ^τ/2 and t∈(λ−εr ^2τ,λ+εr ^2τ)∩I. Then, by the regularity assumption (2.2) and |t−λ|<εr ^2τ, we get

Since the same also holds for all t′ between λ and t, we see that t∈I _j . Finally, suppose that $|\varPhi_{\lambda _{j}}(x,y)| \geq\delta_{I,\tau}r^{\tau}/2$ and fix λ∈(λ _j −εr ^2τ,λ _j +εr ^2τ)∩I. There are three cases. First, if λ∈I _j , then, by choice of λ _j , we clearly have |Φ _λ (x,y)|≥δ _I,τ r ^τ/2. Second, if λ belongs to none of the intervals I _j , we even have the stronger conclusion |Φ _λ (x,y)|≥δ _I,τ r ^τ. Finally, if λ∈I _i for some i≠j, let t∈I be the end point of I _i between λ _j and λ. Then |Φ _t (x,y)|=δ _I,τ r ^τ and |t−λ|≤εr ^2τ so that by (2.2), the mean value theorem, and the choice of ε,

This finishes the proof of (ii). □

Now we are prepared to prove Lemma 4.2. Recall that we should prove

where x,y∈Ω, r=d(x,y), q∈ℕ, j∈ℤ, γ is any compactly supported smooth function on J, and A≥1 is an absolute constant. We may and will assume that q≥2. Moreover, recall that η was a fixed smooth function satisfying $\chi_{[a^{-1},a]} \leq\eta\leq\chi_{[a^{-2},2a]}$ for some large a≥1. Setting $\psi:= \bar{\hat{\eta}}$ (in Appendix A, we freely recycle all the Greek letters and other symbols that had a special meaning in the previous sections), this definition implies that ψ is a rapidly decreasing function, that is, obeys the bounds |ψ(t)|≲ _N (1+|t|)^−N for any N∈ℕ, and also satisfies

$$ \hat{\psi}^{(l)}(0) = 0, \quad l \in \mathbb{N}. $$

(A.1)

Fixing x,y∈Ω, we will also temporarily write

where Φ _λ :=Φ _λ (x,y). For quite some while, it suffices to know that Γ is a smooth function satisfying $\|\varGamma\|_{L^{\infty}(\mathbb{R})} \lesssim_{\gamma} 1$. The inequality we are supposed to prove now takes the form

$$ \biggl \vert \int_{\mathbb{R}} \varGamma ( \lambda)\psi\bigl(2^{j}r\varPhi_{\lambda}\bigr) \, d\lambda\biggr \vert \lesssim_{\gamma,q} \bigl(1 + 2^{j}r^{1 + A\tau} \bigr)^{-q}. $$

(A.2)

We claim that A=14 will do the trick. First of all, we may assume that

$$ 2^{j}r^{1 + m\tau} > 1, \quad0 \leq m \leq14. $$

(A.3)

Indeed, if this were not the case for some 0≤m≤14, then

and we would be done. Next choose an auxiliary function φ∈C ^∞(ℝ) with χ _[−1/4,1/4]≤φ≤χ _[−1/2,1/2]. Then split the integration in (A.2) into two parts:

(A.4)

(A.5)

Here δ _τ :=δ _I,τ >0 is the constant from Definition 2.1, and I⊂J is some compact interval containing the support of Γ. The integral of line (A.5) is easy to bound, since the integrand vanishes whenever |Φ _λ |≤δ _τ r ^τ/4, and if |Φ _λ |≥δ _τ r ^τ/4, we have the estimate

$$ \big|\psi\bigl(2^{j}r\varPhi_{\lambda}\bigr)\big| \lesssim_{q} \bigl(1 + \delta_{\tau}2^{j - 2}r^{\tau+ 1} \bigr)^{-q} \lesssim_{\tau,q} \bigl(1 + 2^{j}r^{\tau+ 1} \bigr)^{-q}, $$

from which

$$ \biggl \vert \int_{\mathbb{R}} \varGamma ( \lambda)\psi\bigl(2^{j}r\varPhi_{\lambda}\bigr)\bigl[1 - \varphi \bigl(\delta_{\tau }^{-1}r^{-\tau}\varPhi_{\lambda} \bigr)\bigr] d\lambda\biggr \vert \lesssim_{q,\gamma,\tau} \bigl(1 + 2^{j}r^{\tau+ 1}\bigr)^{-q}. $$

(A.6)

Moving on to line (A.4), let the intervals I ₁,…,I _N , the points λ _i ∈I _i , and the constant ε>0 be as provided by Lemma A.1 (related to the interval I specified above). Choose another auxiliary function χ∈C ^∞(ℝ) with χ _{[−ε/2,ε/2]}≤χ≤χ _(−ε,ε). Then split the integration on line (A.4) into N+1 parts:

(A.7)

(A.8)

With the aid of part (ii) of Lemma A.1, the integral on line (A.8) is easy to handle. If the integrand is non-vanishing at some point λ∈I, we necessarily have $\varphi(\delta_{\tau}^{-1}r^{-\tau}\varPhi_{\lambda}) \neq0$, which means that |Φ _λ |≤δ _τ r ^τ/2; in particular, λ∈I _i for some 1≤i≤N. Now (a) of Lemma A.1(ii) tells us that (λ−εr ^2τ,λ+εr ^2τ)∩I⊂I _i , from which r ^−2τ|λ−λ _j |≥ε and χ(r ^−2τ(λ−λ _j ))=0 for j≠i. But, since the integrand is non-vanishing at λ∈I, this enables us to conclude that χ(r ^−2τ(λ−λ _i ))<1; in particular, |λ−λ _i |≥εr ^2τ/2. Then Lemma A.1(ii) shows that |Φ _λ |≥δ _τ r ^τ|λ−λ _i |≥εδ _τ r ^3τ/2, and using the rapid decay of ψ as on line (A.6), one obtains

Now we turn our attention to the N integrals on line (A.7). If $|\varPhi_{\lambda_{i}}| \geq\delta_{\tau}r^{\tau}/2$ for some 1≤i≤N, part (b) of Lemma A.1(ii) says that |Φ _λ |≥δ _τ r ^τ/2 for all λ∈(λ _i −εr ^2τ,λ _i +εr ^2τ)∩I, that is, for all λ∈I such that χ(r ^−2τ(λ−λ _i ))≠0. But for such λ∈I it holds that $\varphi(\delta_{\tau}^{-1}r^{-\tau }\varPhi_{\lambda}) = 0$, from which

So we may assume that $|\varPhi_{\lambda_{i}}| < \delta_{\tau}r^{\tau }/2$. In this case, part (a) of Lemma A.1(ii) tells us that the support of λ↦Γ(λ)χ(r ^−2τ(λ−λ _i )) is contained in I _i . The restriction of λ↦Φ _λ to this interval I _i is strictly monotonic, so the inverse $g := \varPhi_{(\cdot)}^{-1} \colon\{\varPhi_{\lambda} : \lambda\in I_{i}\} \to I_{i}$ exists. We perform the change of variables λ↦g(u):

(A.9)

Write $F(u) := \varGamma(g(u))\chi(r^{-2\tau}(g(u) - \lambda_{i}))\varphi(\delta_{\tau}^{-1}r^{-\tau} u)g'(u)$ for u∈{Φ _λ :λ∈I _i }. The support of λ↦Γ(λ)χ(r ^−2τ(λ−λ _i )) is compactly contained in I _i , which means that the support of u↦Γ(g(u))χ(r ^2τ(g(u)−λ _i )) is compactly contained in the open interval {Φ _λ :λ∈I _i }. Thus F may be defined smoothly on the real line by setting F(u):=0 for u∉{Φ _λ :λ∈I _i }. We will need the following lemmas.

Lemma A.2

Let h∈C ^k(a,b) with h′(x)≠0 for x∈(a,b). Suppose the inverse h ⁻¹:h(a,b)→(a,b) exists. Then, for x∈(a,b) and 1≤l≤k,

where the inner summation runs over those b=(b ₁,…,b _m )∈ℕ^m with b ₁+⋯+b _m =(l−1)+m and b _i ≥2 for all 1≤i≤k.

Proof

See [5]. □

Lemma A.3

(Faà di Bruno Formula)

Let f,h∈C ^l(ℝ). Then

$$ (f \circ h)^{(l)}(\lambda) = \sum _{\mathbf{m} \in\mathbb{N}^{l}} b_{\mathbf{m}} f^{(m_{1} + \cdots+ m_{l})}\bigl(h(\lambda) \bigr) \cdot\bigl[h'(\lambda)\bigr]^{m_{1}} \cdots \bigl[h^{(l)}(\lambda)\bigr]^{m_{l}}, $$

(A.10)

where m=(m ₁,…,m _l )∈ℕ^l, and b _m ≠0 if and only if m ₁+2m ₂+⋯+lm _l =l.

Proof

See [6]. □

We apply the first lemma to g. If u=Φ _λ for some λ∈I _i , the derivatives $\partial_{\lambda}^{m}\varPhi_{\lambda}$ satisfy the transversality and regularity conditions of Definition 2.1. Hence

(A.11)

On the last line we simply ignored the summation and employed the inequality

$$ r^{-s} \lesssim_{s,t,d(\varOmega)} r^{-t}, \quad s \leq t. $$

(A.12)

with s=m≤l−1=t (this follows from d(Ω)<∞). Next we wish to estimate the derivatives of F, so we recall that

If u∉{Φ _λ :λ∈I _i }, we have F ^(l)(u)=0 for all l≥0. So, let u=Φ _λ , λ∈I _i . By (A.11) and (A.12) we have |g ^(l)(u)|≲ _l,τ r ^−4τl for l≥1, so (A.10) gives

(A.13)

A similar computation also yields

$$ \big|\partial^{(l)}_{u}\chi \bigl(r^{-2\tau}\bigl(g(u) - \lambda_{i}\bigr)\bigr)\big| \lesssim_{l,\tau} r^{-6\tau l} \quad\text{and} \quad\big| \partial^{(l)}_{u}\varphi \bigl(\delta_{\tau}^{-1}r^{-\tau}u \bigr)\big| \lesssim_{l,\tau} r^{-\tau l}. $$

(A.14)

The presence of the factor $\hat{\eta}(2^{j - i}r\partial_{\lambda}\varPhi_{g(u)})$ in the definition of F(u) is the only place where our proof of Lemma 4.2 differs from the original proof of [15, Lemma 4.6]—and, indeed, we only need to check that the l-th derivatives of this factor admit bounds similar to those of the other factors. Applying (A.10) with $f(\lambda) = \hat{\eta}(2^{j - i}r\partial_{\lambda}\varPhi_{\lambda})$ and h(u)=g(u), we use the bounds |g ^(l)(u)|≲ _l,τ r ^−4τl to obtain

Here, for any k=m ₁+⋯+m _l ≤l and λ∈I, we may use (A.10) again, combined with rapid decay bounds of the form $|\hat{\eta}^{(p)}(t)| \lesssim_{N,p} |t|^{-N}$, to estimate

Finally, recalling that g(u)∈I _i , we have |∂ _λ Φ _g(u)|≳ _τ,I r ^τ by (2.1), from which

and so

Now that we have estimated the derivatives of all the factors of F separately, we may conclude from the Leibniz formula that

$$ \big |F^{(l)}(u)\big| \lesssim_{\gamma ,\tau,l} r^{-7\tau l}, \quad l \geq1. $$

(A.15)

For l=0, estimate (A.11) yields |F(u)|≲ _γ,τ |g′(u)|≲ _τ r ^−τ.

Next we write F as a degree 2(q−1) Taylor polynomial centered at the origin:

Then we split the integration in (A.9) in two for one last time. Denoting U ₁={u∈ℝ:|u|<(2^j r)^−1/2} and U ₂=ℝ∖U ₁,

(A.16)

(A.17)

To estimate the integral of line (A.17) we use the bounds |ψ(2^j ru)|≲ _q (2^j r|u|)^−2q−1, |F(u)|≲ _γ,τ r ^−τ and 2^j r>1 to obtain

(A.18)

As regards (A.16), note that the Fourier transform of x↦x ^l ψ(x) is (a constant multiple of) $i^{l}\hat{\psi }^{(l)}$, from which

according to (A.1). This shows that

The term corresponding to l=0 may be bounded as on line (A.18). For l≥1 we apply the estimates |ψ(2^j ru)|≲ _q (2^j ru)^{−2q−l−1}, 1≤l≤q, and (A.15):

On the last line, we need not actually perform the summation; just note that the terms are decreasing, since 2^j r ^1+7τ>1, and their number is 2(q−1)≲ _q 1. Since also

we have shown that

$$ \Biggl \vert \int_{U_{1}} \psi \bigl(2^{j}r u\bigr) \Biggl[ \sum_{l = 0}^{2(q - 1)} \frac{F^{(l)}(0)}{l!}u^{l} + \mathcal{O} \bigl(F^{(2q - 1)}(u)u^{2q - 1} \bigr) \Biggr] \Biggr \vert \lesssim_{q,\tau} r^{-7\tau(2q - 1)} \bigl(2^{j}r\bigr)^{-q}. $$

(A.19)

Now we wrap things up. On line (A.7) it follows from part (i) of Lemma A.1 that the number N of intervals I _i meeting the support of Γ (or γ) is bounded above by N≲ _γ,τ r ^−2τ. This, together with the estimates (A.18) and (A.19), yields

The last inequality uses assumption 2^j r ^1+14τ≥1. This finishes the proof, since all the finitely many pieces I into which the integral on line (A.2) was decomposed have been seen to satisfy I≲ _{q,γ,τ,d(Ω)}(1+2^j r ^1+cτ)^−q for some 0≤c≤14.