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diff --git a/content/submitted.bib b/content/submitted.bib deleted file mode 100644 index 39b698d..0000000 --- a/content/submitted.bib +++ /dev/null @@ -1,11 +0,0 @@ -@Article{Bukh2018a, - author = {Boris Bukh and Christopher Cox}, - title = {Nearly orthogonal vectors and small antipodal spherical codes}, - abstract = {How can $d+k$ vectors in $\mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $\theta(d,k):=\min_X\max_{x\neq y\in X}|\langle x,y\rangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $X\subseteq\mathbb{R}^d$. In this paper, we focus on the case where $k$ is fixed and $d\to\infty$. In establishing bounds on $\theta(d,k)$, we find an intimate connection to the existence of systems of ${k+1\choose 2}$ equiangular lines in $\mathbb{R}^k$. Using this connection, we are able to pin down $\theta(d,k)$ whenever $k\in\{1,2,3,7,23\}$ and establish asymptotics for general $k$. The main tool is an upper bound on $\mathbb{E}_{x,y\sim\mu}|\langle x,y\rangle|$ whenever $\mu$ is an isotropic probability mass on $\mathbb{R}^k$, which may be of independent interest. Our results translate naturally to the analogous question in $\mathbb{C}^d$. In this case, the question relates to the existence of systems of $k^2$ equiangular lines in $\mathbb{C}^k$, also known as SIC-POVM in physics literature.}, - date = {2018-03-08}, - eprint = {1803.02949v1}, - eprintclass = {math.CO}, - eprinttype = {arXiv}, - file = {online:http\://arxiv.org/pdf/1803.02949v1:PDF}, - keywords = {sub}, -} |