Arithmetic representations of real numbers in terms of self-similar sets

Abstract

Suppose n ≥ 2 and A i ⊂ {0, 1, · · · , (n − 1)} for i = 1, · · · , l, let K i = a∈A i n −1 (K i + a) be self-similar sets contained in [0, 1]. Given m 1 , · · · , m l ∈ Z with i m i = 0, we let Sx = {(y 1 , · · · , y l ) : m 1 y 1 + · · · + m l y l = x with y i ∈ K i ∀i} .In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set Ur = {x : Card(Sx) = r}, where Card(Sx) denotes the cardinality of Sx, and r ∈ N + . We prove under the so-called covering condition that the Hausdorff dimension of U 1 can be calculated in terms of some matrix. Moreover, if r ≥ 2, we also give some sufficient conditions such that the Hausdorff dimension of Ur takes only finite values, and these values can be calculated explicitly. Furthermore, we come up with some sufficient conditions such that the dimensional Hausdorff measure of Ur is infinity. Various examples are provided. Our results can be viewed as the exceptional results for the classical slicing problem in geometric measure theory.2000 Mathematics Subject Classification. Primary 28A80.

Multiple Representations of Real Numbers on Self-Similar Sets With Overlaps

“…1 Athreya, Reznick and Tyson [3] also investigated the division of C, namely the set C ÷ C, and proved that C ÷ C is precisely the countably union of some closed intervals. In [30], Jiang They proved that dim H (U r ) = log 2 log 3 if r = 2 k for some k ∈ N. Moreover,…”

“…U 3·2 k is an infinitely countable set for any k ≥ 1, where dim H and H s denote the Hausdorff dimension and Hausdorff measure, respectively. There are more general results in [30]. In this paper, we shall analyze the following self-similar set with overlaps [26], let K be the attractor generated by the following IFS,…”

Arithmetic on Moran Sets

“…U 3·2 k is an infinitely countable set for any k ≥ 1, where dim H and H s denote the Hausdorff dimension and Hausdorff measure, respectively. For more results, see [14]. In [25], Tian et al defined a class of overlapping self-similar sets as follows: let K be the attractor of the IFS…”

“…Jiang and Xi[13] proved that C • C indeed contains infinitely many intervals. In[14], Jiang and Xi considered the representations of real numbers in C…”

Interiors of continuous images of self-similar sets with overlaps

“…These representations are popular in number theory. There are, however, some other approaches that can represent real numbers ( [2], [24], [37], [39]). First, let us introduce some basic definitions.…”