Optimal expansions in non-integer bases

“…A multinacci number is the unique root of an equation of the form x n+1 = x n + · · · + x + 1 contained in (1, 2). The main result of [4] is the following. Theorem 1.9.…”

Section: Introductionmentioning

confidence: 99%

“…The approximation properties of β-expansions were also studied by Dajani, Komornik, Loreti, and de Vries in [4]. Given x ∈ I β , they call a sequence…”

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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“…Then b p (x) := (b i ) is an expansion of x in base p. 2 By construction this is the lexicographically largest expansion of x in base p, called the greedy or β-expansion of x in base p. Today there is a huge literature devoted to non-integer expansions. For example, probabilistic and ergodic aspects are investigated in [3], [5], [31], [32], [34], combinatorial properties in [1], [4], [18], [25], [26], [30], unique expansions in [6], [7], [8], [9], [10], [12], [14], [15], [16], [21], [22], [23], [24], [27], and control-theoretical applications are given in [2], [28], [29].…”

Section: Introductionmentioning

confidence: 99%