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Thomas Ramsey","submitted_at":"2015-03-26T20:33:35Z","abstract_excerpt":"For any three element set of positive integers, $\\{a,b,n\\}$, with $a<b<n$, $n$ sufficiently large and $\\gcd(a,b)=1$, we find the least $\\alpha$ such that given any real numbers $t_1$, $t_2$, $t_3$, there is a real number $x$ such that \\begin{equation*} \\max \\{\\left\\langle ax-t_{1}\\right\\rangle ,\\left\\langle bx-t_{2}\\right\\rangle ,\\left\\langle nx-t_{3}\\right\\rangle \\}\\leq \\alpha , \\end{equation*} where $\\left\\langle \\cdot \\right\\rangle $ denotes the distance to the nearest integer. The number $\\alpha $ is known as the angular Kronecker constant of $\\{a,b,n\\}$. 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