Over the next couple of weeks, I will be posting short expositions of various parts of an upcoming paper by Malabika Pramanik and myself on maximal estimates associated with sparse sets in

The following question was raised and investigated by Vincenzo Aversa and David Preiss in the 1980s and 90s: to what extent can the Lebesgue density theorem be viewed as “canonical” in

Let’s make this more precise. For the purpose of this post, we will say that family *density property* if for every measurable set

This is slightly different from standard terminology, but there should be no danger of confusion, as we will not use any other density properties here. We write

The Lebesgue density theorem states that the collection of intervals

But that does not exhaust all examples. For instance, consider the family *hearts density theorem* of Preiss and Aversa-Preiss.

Note, however, that the collection in the last example is not closed under scaling *intervals* is invariant under such scaling and has the density property, then its density property must follow from the Lebesgue theorem in the manner described above.

On the other hand, if we consider more general sets than intervals, then it turns out that there are indeed scaling-invariant density theorems that are independent of the Lebesgue theorem. This was announced by Aversa and Preiss in 1987; the proof (via a probabilistic construction) was published in a 1995 preprint.

Theorem 1 (Aversa-Preiss):There is a sequenceof compact sets of positive measure such that and:

is a Lebesgue density point for , and in particular we have - the family
has the density property.

The analogous question for

We will say that *differentiates*

For example, the Lebesgue differentiation theorem states that the collection

The differentiation property is formally stronger than the density property, by letting

The following theorem was conjectured by Aversa and Preiss in 1995, and proved very recently by Malabika Pramanik and myself (paper in preparation).

Theorem 2.There is a sequenceof compact sets of positive measure with such that differentiates . More explicitly, for every we have

for a.e.

Our construction of *maximal operator* associated with it is bounded on appropriate

The exact statement of the maximal estimate, and some of the ideas from the proof, will follow in the next installment.