Approximation properties of β-expansions

Baker, Simon

doi:10.4064/aa168-3-4

Cited by 10 publications

(21 citation statements)

References 27 publications

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“…As such, there is a dense set of exceptions to Conjecture 1.1. In [1] we were able to verify Conjecture 1.1 for a class of algebraic integers known as Garsia numbers. These numbers are the positive real algebraic integers with norm ±2, whose conjugates are all of modulus strictly greater than 1.…”

Section: Introductionmentioning

confidence: 95%

“…Similarly, if the level n sums are well distributed within I β , we would expect to have good approximation properties. This behaviour was observed in [1]. Understanding how the level n sums are distributed within I β is a long standing and classical problem.…”

Section: Introductionmentioning

confidence: 97%

“…In (1.2) the approximation to x is given by a level n sum, not necessarily an n prefix for x. However, as is explained in [1], if (1.2) is satisfied by an arbitrary ( i ) n i=1 , then it is also satisfied by an n-prefix of x.…”

Section: Introductionmentioning

confidence: 99%

“…These numbers are the positive real algebraic integers with norm ±2, whose conjugates are all of modulus strictly greater than 1. In particular the main result of [1] is the following. Note that Garsia showed that whenever β is a Garsia number, the associate Bernoulli convolution µ β is absolutely continuous with respect to the Lebesgue measure, and has bounded density [8].…”

Section: Introductionmentioning

confidence: 99%

“…Note that if ∞ n=1 2 n Ψ(n) < ∞, then W β (Ψ) will always have zero Lebesgue measure by the Borel-Cantelli lemma. For a generic β we expect the level n sums to be well distributed within I β , as such we made the following conjecture in [1]. Conjecture 1.1.…”

Section: Introductionmentioning

confidence: 99%

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Approximation properties of -expansions II

Baker¹

2017

Ergod. Th. Dynam. Sys.

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Let $\unicode[STIX]{x1D6FD}\in (1,2)$ be a real number. For a function $\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{\geq 0}$, define $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ to be the set of $x\in \mathbb{R}$ such that for infinitely many $n\in \mathbb{N},$ there exists a sequence $(\unicode[STIX]{x1D716}_{i})_{i=1}^{n}\in \{0,1\}^{n}$ satisfying $0\leq x-\sum _{i=1}^{n}(\unicode[STIX]{x1D716}_{i}/\unicode[STIX]{x1D6FD}^{i})\leq \unicode[STIX]{x1D6F9}(n)$. In Baker [Approximation properties of $\unicode[STIX]{x1D6FD}$-expansions. Acta Arith. 168 (2015), 269–287], the author conjectured that for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1,2)$, the condition $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)=\infty$ implies that $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. In this paper we make a significant step towards proving this conjecture. We prove that given a sequence of positive real numbers $(\unicode[STIX]{x1D714}_{n})_{n=1}^{\infty }$ satisfying $\lim _{n\rightarrow \infty }\unicode[STIX]{x1D714}_{n}=\infty$, for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1.497,\ldots ,2)$, the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D714}_{n}\cdot 2^{-n})$ is of full Lebesgue measure within $[0,1/(\unicode[STIX]{x1D6FD}-1)]$. We also study the case where $\sum _{n=1}^{\infty }2^{n}\unicode[STIX]{x1D6F9}(n)<\infty$ in which the set $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$ has Lebesgue measure zero. Applying the mass transference principle developed by Beresnevich and Velani in [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], we obtain some results on the Hausdorff dimension and the Hausdorff measure of $W_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D6F9})$.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximation properties of -expansions II

Baker¹

2017

Ergod. Th. Dynam. Sys.

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Intrinsic Diophantine approximation for overlapping iterated function systems

Baker

2023

Math. Ann.

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In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchine’s theorem for these sets. We then apply this result to the topic of intrinsic Diophantine Approximation on self-similar sets. In particular, we define a new height function for an element of $${\mathbb {Q}}^d$$ Q d contained in a self-similar set in terms of its eventually periodic representations. For limsup sets defined with respect to this height function, we obtain a detailed description of their metric properties. The results of this paper hold in arbitrary dimensions and without any separation conditions on the underlying iterated function system.

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