arXiv subjects
Publications and source records attributed to Amy Woodall.
We introduce and study arithmetic polygons. We show that these arithmetic polygons are connected to triples of square pyramidal numbers. For every odd $N\geq3$, we prove that there is at least one arithmetic polygon with $N$ sides. We also show that there are infinitely many arithmetic polygons with an even number of sides.
Let $L$ be an even lattice of odd rank with discriminant group $L'/L$, and let $α,β\in L'/L$. We prove the Weil bound for the Kloosterman sums $S_{α,β}(m,n,c)$ of half-integral weight for the Weil Representation attached to $L$. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen's identity for plus space Kloosterman sums with the theta multiplier system.
In recent work, the first two authors constructed a generalized continued fraction called the $p$-continued fraction, characterized by the property that its convergents (a subsequence of the regular convergents) are best approximations with respect to the $L^p$ norm, where $p\geq 1$. We extend this construction to the region $0<p<1$, where now the $L^p$ quasinorm is non-convex. We prove that the approximation coefficients of the $p$-continued fraction are bounded above by $1/\sqrt{5}+\varepsilon_p$, where $\varepsilon_p\to 0$ as $p\to 0$. In light of Hurwitz's theorem, this upper bound is sharp, in the limit. We also measure the maximum number of consecutive regular convergents that are skipped by the $p$-continued fraction.