M. Bricchi

Existence and Properties of h-Sets

We shall consider the following problem: which conditions should satisfy a function h : --> R in order to guarantee the existence of a (regular) measure m in Rn with compact subset G of Rn as support and
(*)    c1h(r) <= m (B(g, r)) <= c2 h(r),
for some positive constants c1, and c2 independent of g in G and r, 0 < r < 1? The theory of self-similar fractals provides outstanding examples of sets fulfilling (*) with h(r) = rd, 0 <= d <= n, and a suitable measure m. Analogously, we shall rely on some recent techniques for the construction of pseudo self-similar fractals in order to deal with our more general task.