Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable

Abstract

Abstract. A long-standing conjecture of Lapidus claims that under certain conditions, self-similar fractal sets fail to be Minkowski measurable if and only if they are of lattice type. The theorem was established for fractal subsets of R by Falconer, Lapidus and v. Frankenhuijsen, and the forward direction was shown for fractal subsets of R d , d ≥ 2, by Gatzouras. Since then, much effort has been made to prove the converse. In this paper, we prove a partial converse by means of renewal theory. Our proof allows us to recover several previous results in this regard, but is much shorter and extends to a more general setting; several technical conditions appearing in previous versions of this result have now been removed.

Lattice self-similar sets on the real line are not Minkowski measurable

“…Our main results are stated in Section 3 and proved in Section 4. We conclude by showing in Section 5 that for sets in R the above-mentioned results from [KK15,KPW16] are equivalent.…”

“…In the proof of Theorem 3.1 we will make use of a general Minkowski measurability criterion for self-similar sets in R d (satisfying OSC) derived in [KPW16]. It is based on feasible open sets satisfying the projection condition and was obtained via classical renewal theory.…”

“…for some m ∈ N 0 , then F is not Minkowski measurable. (v) [KPW16] Assume existence of a strong feasible open set O for Φ that satisfies the projection condition and that allows for a finite partition of (0, ∞)…”

“…In the present article we fully remove the assumptions of [LvF00, KK15,KPW16] and in this way provide the last puzzle-piece in proving that under OSC a nontrivial self-similar subset of R is Minkowski measurable iff it arises from a nonlattice IFS. This resolves the conjecture stated in [Lap93, Conjecture 3] and [Gat00, Section 5.2] for self-similar sets in R.…”

“…For nontrivial selfsimilar subsets of R the converse, i. e. lattice sets are not Minkowski measurable, was shown in [LvF00] under additional assumptions. These assumptions address the geometric structure of the underlying feasible open set for the OSC and have been weakened in [KK15,KPW16], see Section 2.3 for more details. However, up to now the conjecture remained unresolved for large classes of self-similar sets, see Section 2.4 for examples.…”