Discrete–geometric functions associated to polyhedral cones and point sets

Abstract

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 limǫ→0+ X (m,n)∈Z2 m(cm+dn)6=0 e−πǫ(m2+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) ∈ Z2 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 integral cones. We managed to generalize our argument from integral cones to real cones. We studied the convergent property of the sum: X(m,n)∈Z2 (αm+n)(m+βn)6=0 f(ǫ, α, β) = X (m,n)∈Z2 (αm+n)(m+βn)6=0 e−πǫ(m2+n2) (αm + n)(m + βn), which is defined on real cones. Our conclusion is when α and β are both quadratic irrationals, this infinite series will converge absolutely. Meanwhile we also gave a sufficient condition for the existence of the limit: limǫ→0+ f(ǫ, α, β). In the following two chapters, our main interest lies in Dedekind sums. In Chapter 4, we focus on the classical Dedekind sum: s(c, d) = d−1 X k=0 kcd kd and answer the question of when two Dedekind sums are equal to each other. We have found a necessary condition which is b|(1 − a1a2)(a1 − a2) in order for s(a1, b) to be equal to s(a2, b). A parallel analysis for the Dedekind-Rademacher sum, namely rn(a, b) = b−1 X k=0 ka + n bkb, is also given in Chapter 4. We include part of the content from this chapter in our paper [JRW11]. In Chapter 5, our focus is on Zagier-Dedekind sums, or higher dimensional Dedekind sums: d(p; a1, · · · , an) = (−1)n/2 p−1 X k=1 cot πka1 p · · · cot πkanp, where p is a positive integer, a1, · · · , an are integers relatively prime to p and n even. The condition for two Zagier-Dedekind sums to be equal to each other is slightly more complicated than the one we gave for classical Dedekind sums. An interesting fact about Zagier-Dedekind sums is that there is a nice relation between d(p; a1, · · · , an) and counting lattice points whose definition depends on a1, a2, · · · , an and p. In Chapter 6, we study a curve which is defined as a set of generalized centers of a finite point set. We call it μ-curve for short. This curve is infinitely smooth, and it captures the symmetrical properties of the original point set such as radial symmetry, reflectional symmetry, and rotational symmetry. We generalize Weiszfeld’s algorithm to find μ(r) through an iteration process for r ≥ 1. We prove that the μ-curve is invariant under rigid motions, and we conjecture that the nondegenerate μ-curve is uniquely determined by a point set. An example is given to support this conjecture. We also give plenty of examples of the μ-curve for different point sets in the end of this chapter.