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In this report, we use Fourier analysis and Diophantine analysis to study functions associated to polyhedral cones and finite point sets. We present a close relationship between the function associated to integral cones and the classical Dedekind sums. The theory of the polytope algebra—the universal group for translation-invariant valuations—was developed by many mathematicians (see [MS83], [Bri97]). In this report, we employed forms of evaluation, namely rational function valuation, which lead us to Dedekind sums. The report is constructed as follows. The first chapter is served as an introduction to the whole thesis. In Chapter 2, we consider the decomposition of the first quadrant cone into integral cones and study the asymptotic behavior of an infinite sum: f(c, d) = 1 4π2 lim ǫ→0+ X (m,n)∈Z2 m(cm+dn)6=0 e−πǫ(m 2+n2) m(cm + dn) . The motivation for us to study this function arises from the Fourier transform of indicator function of cones. Decomposition of the first quadrant cone leads to an identity: the indicator function of the first quadrant cone is equal to the sum of indicator functions of the two cones after decomposition. In order to apply harmonic analysis in our work, we first smooth out indicator function of cones by Gaussian function. By doing so we may apply certain identities such as the Poisson Summation Formula to the modified indicator function. These facts combined grant us to explore the limit of an infinite sum f(c, d) where the integral vector (c, d) ∈ Z is the common edge shared by two cones after decomposition of the first quadrant cone. In the end of Chapter 2, we discovered a nice relationship between f(c, d) and the classical Dedekind sum s(c, d) which is the main research object of Chapter 4. In Chapter 3, we continue to use a similar technique which appeared earlier in Chapter 2 to investigate an infinite sum defined over cones. The main difference is here we focus on real cones while earlier in Chapter 2, we are interested in

Discrete–geometric functions associated to polyhedral cones and point sets