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Register a new userCyclic shifts of the van der Corput set
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Authors
Dmitriy Bilyk
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In [13], K. Roth showed that the expected value of the $L^2$ discrepancy of the cyclic shifts of the $N$ point van der Corput set is bounded by a constant multiple of $sqrt{log N}$, thus guaranteeing the existence of a shift with asymptotically minimal $L^2$ discrepancy, [11]. In the present paper, we construct a specific example of such a shift.
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The van der Waals heterostructures are a fertile frontier for discovering emergent phenomena in condensed matter systems. They are constructed by stacking elements of a large library of two-dimensional materials, which couple together through van der Waals interactions. However, the number of possible combinations within this library is staggering, and fully exploring their potential is a daunting task. Here we introduce van der Waals metamaterials to rapidly prototype and screen their quantum counterparts. These layered metamaterials are designed to reshape the flow of ultrasound to mimic electron motion. In particular, we show how to construct analogues of all stacking configurations of bilayer and trilayer graphene through the use of interlayer membranes that emulate van der Waals interactions. By changing the membranes density and thickness, we reach coupling regimes far beyond that of conventional graphene. We anticipate that van der Waals metamaterials will explore, extend, and inform future electronic devices. Equally, they allow the transfer of useful electronic behavior to acoustic systems, such as flat bands in magic-angle twisted bilayer graphene, which may aid the development of super-resolution ultrasound imagers.
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Let $G$ be a finite abelian group. We say that $M$ and $S$ form a textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $min M$ and $sin S$, while $0$ has no such representation. The splitting is called textit{purely singular} if for each prime divisor $p$ of $|G|$, there is at least one element of $M$ is divisible by $p$. In this paper, we mainly study the purely singular splittings of cyclic groups. We first prove that if $kge3$ is a positive integer such that $[-k+1, ,k]^*$ splits a cyclic group $mathbb{Z}_m$, then $m=2k$. Next, we have the following general result. Suppose $M=[-k_1, ,k_2]^*$ splits $mathbb{Z}_{n(k_1+k_2)+1}$ with $1leq k_1< k_2$. If $ngeq 2$, then $k_1leq n-2$ and $k_2leq 2n-5$. Applying this result, we prove that if $M=[-k_1, ,k_2]^*$ splits $mathbb{Z}_m$ purely singularly, and either $(i)$ $gcd(s, ,m)=1$ for all $sin S$ or $(ii)$ $m=2^{alpha}p^{beta}$ or $2^{alpha}p_1p_2$ with $alphageq 0$, $betageq 1$ and $p$, $p_1$, $p_2$ odd primes, then $m=k_1+k_2+1$ or $k_1=0$ and $m=k_2+1$ or $2k_2+1$.
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