Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 1111-1129

ARITHMETIC REPRESENTATIONS OF REAL NUMBERS IN TERMS OF SELF-SIMILAR SETS

Kan Jiang and Lifeng Xi

Ningbo University, Department of Mathematics
Ningbo 315211, P.R. China; jiangkan 'at' nbu.edu.cn

Ningbo University, Department of Mathematics
Ningbo 315211, P.R. China; xilifeng 'at' nbu.edu.cn

Abstract. Suppose n ≥ 2 and A_i ⊂ {0,1,...,(n – 1)} for i = 1,...,l, let K_i = ∪_a &isin A_in^-1(K_i + a) be self-similar sets contained in [0,1]. Given m₁,...,m_l ∈ Z with ∏_im_i ≠ 0, we let

S_x = {(y₁,...,y_l) : m₁y₁ + ... + m_ly_l = x with y_i ∈ K_i ∀ i}.

In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set

U_r ={x : Card(S_x) = r},

where Card(S_x) denotes the cardinality of S_x, and r ∈ N⁺. We prove under the so-called covering condition that the Hausdorff dimension of U₁ can be calculated in terms of some matrix. Moreover, if r ≥ 2, we also give some sufficient conditions such that the Hausdorff dimension of U_r 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 U_r is infinity. Various examples are provided. Our results can be viewed as the exceptional results for the classical slicing problem in geometric measure theory.

2010 Mathematics Subject Classification: Primary 28A80.

Key words: Fractal, self-similar set, unique representation, section, projection.

Reference to this article: K. Jiang and L. Xi: Arithmetic representations of real numbers in terms of self-similar sets. Ann. Acad. Sci. Fenn. Math. 44 (2019), 1111-1129.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4463