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We explore lattice point counting and the method of stationary phase through the lens of questions about the number of lattice points on and near surfaces with vanishing curvature. Our focus is on spheres arising from the Heisenberg groups. In particular, we prove an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic Heisenberg norms for α ≥ 2. We accomplish this through a transformative process that takes a number theory question about counting lattice points and translates it into that of an analytical estimation of measure. This process relies on truncating and scaling the n-dimensional integer lattice to produce a fractal-like set. By introducing a measure on this resulting set and using elementary Fourier analysis, the counting problem is transformed into one of bounding an energy integral.This process uses principles of fractal geometry and oscillatory integrals. Primary challenges that arise are the presence of vanishing curvature and uneven dilations. Following a discussion and formal estimate of the curvature of the Heisenberg spheres, we utilize the method of stationary phase to compute a bound on the Fourier transform of their surface measures.Our work is inspired by that of Iosevich and Taylor (2011) and Garg, Nevo, and Taylor (2015). We present an extension of the main result in the former to surfaces with vanishing curvature. Furthermore, we utilize the techniques developed here to estimate the number of lattice points in the intersection of two such surfaces.Additionally, we present a mini-course on the basics of stationary phase—a quick-start guide to stationary phase in practice. This includes a discussion of the formulation of oscillatory integrals and their solutions with a focus on the impact of geometric properties (e.g. curvature) on the estimates for the decay of the Fourier transform. It further serves as a supplement to [Shakarchi and Stein, Functional Analysis: Chapter 8, Sections 2 and 3], providing detailed expositions of the proofs of fundamental results.We conclude with a discussion of potential avenues for improvement and interesting problems that could be considered utilizing our methods.

The purpose of this study was to show how mathematics can be used to analyze effects on sound. Our hope is that this may inspire student interest in mathematics.We analyzed five common industry standard effects. Research data was gathered using Mathematica and GarageBand software. Three versions of each effect were used to alter pure tone sound waves of ten different frequencies using GarageBand. Then using Mathematica's Fourier command, the frequency spectrum of each altered sound wave was generated. Through observation of each set of 30 frequency spectra, the most prominent and common pure tone components were determined. For each effect, Mathematica's Fit command was used to determine a best fit model of the magnitude of each component as a function of frequency. Our models provide descriptions of the effects that are consistent with the traditional descriptions of the industry standard effects in our study. If similar research is to be conducted, our recommendation is that more versions of each effect, a wider range of input frequencies, and a higher sampling rate would produce function models that are even more consistent with traditionally accepted effect descriptions. Furthermore, an understanding of the hardware and software design used to build effects on sound is highly recommended.