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Let $\mathbf{A}=\{A_i\}_{i=1}^{\infty}$ be a sequence of sets with each $A_i$ being a non-empty collection of $0$-$1$ sequences of length $i$. For $x\in [0,1)$, the maximal run-length function $\ell_n(x,\mathbf{A})$ (with respect to $\mathbf{A}$) is defined to the largest $k$ such that in the first $n$ digits of the dyadic expansion of $x$ there is a consecutive subsequence contained in $A_k$. Suppose that $\lim_{n\to\infty}(\log_2|A_n|)/n=\tau$ for some $\tau\in [0,1]$ and one additional assumption holds, we prove a generalization of the Erd\H{o}s-R\'enyi limit theorem which states that \[\lim_{n\to\infty}\frac{\ell_n(x,\mathbf{A})}{\log_2n}=\frac{1}{1-\tau}\] for Lebesgue almost all $x\in [0,1)$. For the exceptional sets, we prove under a certain stronger assumption on $\mathbf{A}$ that the set \[\left\{x\in [0,1): \lim_{n\to\infty}\frac{\ell_n(x,\mathbf{A})}{\log_2n}=0\text{ and } \lim_{n\to\infty}\ell_n(x,\mathbf{A})=\infty\right\}\] has Hausdorff dimension at least $1-\tau$.

We prove that for two connected sets $E,F\subset\mathbb{R}^2$ with cardinalities greater than $1$, if one of $E$ and $F$ is compact and not a line segment, then the arithmetic sum $E+F$ has non-empty interior. This improves a recent result of Banakh, Jab{\l}o\'nska and Jab{\l}o\'nski [4,Theorem 4] in dimension two by relaxing their assumption that $E$ and $F$ are both compact., Comment: Accepted for publicaton in Bull. Aust. Math. Soc

In this paper, we investigate inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems. For $\beta>1$ let $T_{\beta}$ be the $\beta$-transformation on $[0,1]$. We determine the Lebesgue measure and Hausdorff dimension of the set \[\left\{(x,y)\in [0,1]^2: |T_{\beta}^nx-f(x,y)|<\varphi(n)\text{ for infinitely many }n\in\mathbb{N}\right\},\] where $f:[0,1]^2\to [0,1]$ is a Lipschitz function and $\varphi$ is a positive function on $\mathbb{N}$. Let $\beta_2\geq \beta_1>1$, $f_1,f_2:[0,1]\to [0,1]$ be two Lipschitz functions, $\tau_1,\tau_2$ be two positive continuous functions on $[0,1]$. We also determine the Hausdorff dimension of the set \[\left\{(x,y)\in [0,1]^2: \begin{aligned}&|T_{\beta_1}^nx-f_1(x)|<\beta_1^{-n\tau_1(x)}\\ &|T_{\beta_2}^ny-f_2(y)|<\beta_2^{-n\tau_2(y)}\end{aligned}\text{ for infinitely many }n\in\mathbb{N}\right\}.\] Under certain additional assumptions, the Hausdorff dimension of the set \[\left\{(x,y)\in [0,1]^2: \begin{aligned}&|T_{\beta_1}^nx-g_1(x,y)|<\beta_1^{-n\tau_1(x)}\\ &|T_{\beta_2}^ny-g_2(x,y)|<\beta_2^{-n\tau_2(y)}\end{aligned}\text{ for infinitely many }n\in\mathbb{N}\right\}\] is also determined, where $g_1,g_2:[0,1]^2\to [0,1]$ are two Lipschitz functions.

We propose a quantum repeater architecture that can operate under ambient conditions. Our proposal builds on recent progress towards non-cryogenic spin-photon interfaces based on nitrogen-vacancy centers, which have excellent spin coherence times even at room temperature, and optomechanics, which allows to avoid phonon-related decoherence and also allows the emitted photons to be in the telecom band. We apply the photon number decomposition method to quantify the fidelity and the efficiency of entanglement established between two remote electron spins. We describe how the entanglement can be stored in nuclear spins and extended to long distances via quasi-deterministic entanglement swapping operations involving the electron and nuclear spins. We furthermore propose schemes to achieve high-fidelity readout of the spin states at room temperature using the spin-optomechanics interface. Our work shows that long-distance quantum networks made of solid-state components that operate at room temperature are within reach of current technological capabilities.

We prove that for all $s\in(0,d)$ and $c\in (0,1)$ there exists a self-similar set $E\subset \mathbb{R}^d$ with Hausdorff dimension $s$ such that $\mathcal{H}^s(E)=c|E|^s$. This answers a question raised by Zhiying Wen[16].

We establish matrix representations for self-similar measures on $\mathbb{R}^d$ generated by equicontractive IFSs satisfying the finite type condition. As an application, we prove that the $L^q$-spectrum of every such self-similar measure is differentiable on $(0,\infty)$. This extends an earlier result of Feng (J. Lond. Math. Soc.(2) 68(1):102--118, 2003) to higher dimensions., Comment: Accepted for publication in Mathematische Zeitschrift

We propose a quantum repeater architecture that can operate under ambient conditions. Our proposal builds on recent progress towards non-cryogenic spin-photon interfaces based on nitrogen-vacancy centers, which have excellent spin coherence times even at room temperature, and optomechanics, which allows to avoid phonon-related decoherence and also allows the emitted photons to be in the telecom band. We apply the photon number decomposition method to quantify the fidelity and the efficiency of entanglement established between two remote electron spins. We describe how the entanglement can be stored in nuclear spins and extended to long distances via quasi-deterministic entanglement swapping operations involving the electron and nuclear spins. We furthermore propose schemes to achieve high-fidelity readout of the spin states at room temperature using the spin-optomechanics interface. Our work shows that long-distance quantum networks made of solid-state components that operate at room temperature are within reach of current technological capabilities., Comment: This paper has been updated with the identifier: arXiv:2203.06611 which has been published in Quantum as Quantum 6, 669 (2022) with the related DOI: https://doi.org/10.22331/q-2022-03-17-669

A compact set $E\subset {\Bbb R}^d$ is said to be arithmetically thick if there exists a positive integer $n$ so that the $n$-fold arithmetic sum of $E$ has non-empty interior. We prove the arithmetic thickness of $E$, if $E$ is uniformly non-flat, in the sense that there exists $\epsilon_0>0$ such that for $x\in E$ and $0

We analyze and compare three different schemes that can be used to generate entanglement between spin qubits in optically-active single solid-state quantum systems. Each scheme is based on first generating entanglement between the spin degree of freedom and either the photon number, the time bin, or the polarization degree of freedom of photons emitted by the systems. We compute the time evolution of the entanglement generation process by decomposing the dynamics of a Markovian master equation into a set of propagation superoperators conditioned on the cumulative detector photon count. We then use the conditional density operator solutions to compute the efficiency and fidelity of the final spin-spin entangled state while accounting for spin decoherence, optical pure dephasing, spectral diffusion, photon loss, phase errors, detector dark counts, and detector photon number resolution limitations. We find that the limit to fidelity for each scheme is restricted by the mean wavepacket overlap of photons from each source, but that these bounds are different for each scheme. We also compare the performance of each scheme as a function of the distance between spin qubits., Comment: 16 pages, 8 figures