Density and ball bounds for Mattila's Lemma 14.7 #
This file develops the quantitative density and ball-measure estimates used in Lemma 14.7.
Densities #
For a measure on Euclidean space, the upper s-density dimensionalUpperDensity
(from DensitiesBasic) is the limsup as r ↓ 0 of μ (closedBall a r) / (2 r) ^ s.
For a measure on Euclidean space, the lower s-density dimensionalLowerDensity
(from DensitiesBasic) is the liminf as r ↓ 0 of μ (closedBall a r) / (2 r) ^ s.
The set A of Mattila, Lemma 14.7: the points a at which
0 < Θ^s_*(μ, a) ≤ Θ^{*s}(μ, a) < ∞.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The density ratio t (a) = Θ^s_*(μ, a) / Θ^{*s}(μ, a) of Mattila, Lemma 14.7: the lower
s-dimensional density of μ at a divided by the upper one.
Equations
- RatioOfDensities s μ a = dimensionalLowerDensity μ.toOuterMeasure s a / dimensionalUpperDensity μ.toOuterMeasure s a
Instances For
The quantity limsup_{δ ↓ 0} sup {d (B) ^ (-s) μ (B) : B a closed ball with z ∈ B and d (B) < δ} appearing in the hypothesis of Mattila, Lemma 14.7 (2). Here d (B) = 2 ρ is the
diameter of the ball B = B (y, ρ).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Supports #
A set all of whose points lie outside the support of μ is μ-null.
(Second countability of the space is what makes the usual Lindelöf argument work.)
An open set of positive measure contains a point of the support.
Blow-ups of balls with arbitrary centres #
Fubini bound: the integral over x ∈ B (x₀, R) of ν (B (x, r)) is at most r ^ n times
the volume of the unit ball times ν (B (x₀, R + r)).
A locally finite Borel measure on ℝⁿ cannot satisfy a uniform lower bound
c ρ ^ s ≤ ν (B (x, ρ)) at every point when s < n and c > 0.
The support of a measure with s-dimensional lower growth, s < n, is not everything.
If ν is a Radon measure (a regular Borel measure) on ℝⁿ and there is c > 0 with
c ρ ^ s ≤ ν (B (x, ρ)) for every x ∈ spt ν and every ρ > 0, and if s < n, then
spt ν ≠ ℝⁿ. This is the step of Mattila's proof of Lemma 14.7 (3) which produces a point
outside the support of the tangent measure supplied by part (1).
Holes at every scale.
If s < n and the measures of the balls centred on F of radius ρ < r₀ are between
p ρ ^ s and q ρ ^ s, then there is κ > 0 such that every ball B (x, δ) with x ∈ F and
δ < r₀ / 2 contains a point at distance more than κ δ from F.
A few elementary facts #
A nonzero measure charges some ball centred at the origin.
The engine behind Lemma 14.7 (4) #
If x belongs to the support of a blow-up limit ν, then, for large i, the point
a + r i • x is within distance ε * r i of the support of μ.
Upper bound in Lemma 14.7 (1), from the uniform upper bound on spt μ.
Lower bound in Lemma 14.7 (1), from the uniform lower bound on spt μ.
The scaling constants of a blow-up sequence are bounded, under the uniform lower bound.
The scaling constants of a blow-up sequence do not degenerate, under the uniform upper bound.
The doubling condition from uniform ball bounds #
Uniform two-sided ball bounds on spt μ imply Mattila's assumption 14.3 (1) at every point
of spt μ.
Nearby points of a set with a density point #
This is the ingredient which replaces, in the proof of Lemma 14.7 (1), the elementary fact used
for Lemma 14.7 (4) that an open set of positive measure meets spt μ.
If a is a μ-density point of a set B on which the measures of small balls are
comparable to ρ ^ s, and x lies in the support of a blow-up limit ν of μ at a, then for
large i the point a + r i • x is within distance ε * r i of B.
Under the hypotheses of the density-point lemma, the blow-ups give arbitrarily small mass to
sets which live at a fixed multiple of the scale r i around a and avoid B.
Upper bound for tangent measures of balls with arbitrary centres, from a bound on the
measures of all small balls meeting the set P.
Mattila's reduction step for Lemma 14.7 (1) #
Mattila, Lemma 14.7, reduction step.
Let τ, d, r₀ be positive numbers and let B be a set with
τ d ρ ^ s ≤ μ (B (y, ρ)) ≤ d ρ ^ s for y ∈ B and 0 < ρ < r₀.
If a ∈ B is a μ-density point of B, then every tangent measure ν ∈ Tan (μ, a) satisfies
τ c ρ ^ s ≤ ν (B (x, ρ)) ≤ c ρ ^ s for x ∈ spt ν and 0 < ρ < ∞, for some positive finite
constant c.
Mattila's reduction step for Lemma 14.7 (2) #
Mattila, Lemma 14.7, reduction step for part (2).
If, in addition to the hypotheses of the reduction step for part (1), the measures of all small
balls meeting B are bounded by d ρ ^ s, then the upper bound for the tangent measure holds for
balls with arbitrary centres, with the same constant c.