On the embedding of triangles into integer lattices
Blair, James David
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An important question in Number Theory has been ‘When can a positive integer l be written as the sum of n squares?’ This question has been answered and refined by Gauss and Jacobi, amongst many others. To generalize, consider the geometric version: ‘Are there vectors of length pl in Zn?’ Then it is clear that there is a natural progression from embedding vectors, to triangles and higher dimensional simplices. We start by analyzing what must be true if a triangle is realizable in Zn. This yields some simple requirements; however the difficulty lies in finding sufficient conditions. Computational evidence suggests a 2-adic condition in 4 dimensions, while the neccessary conditions are sufficient for n > 5. Studying n = 4 for a local-global principle requires solving a system of equations p-adically for all primes. This is possible for p > 2; furthermore, a solution when p = 2 may be used to generate a global solution via the ring of Integer Quaternions. From there it follows that every admissible triangle sits in Z5. The next natural problem is to estimate how many ways a triangle may be embedded. When n = 4, this uses a result of Siegel’s that a number l may be written as the sum of three squares in C l 1 2- ways. For n = 5, methods of projecting down a dimension are computed, and then Siegel is applied again. A more analytic approach is then taken, generalizing the Circle Method. This amounts to considering rational scalings of the triangle, and provides an averaged version of the estimate. Future development is suggested in two ways. First, higher dimensional simplices are more difficult to embed since many of the tools for triangles depend on the Quaternions. Analysis is still possible for odd primes p, suggesting a 2-adic restriction again. Finally, by combining results with estimates for Fourier coefficients of modular forms (work being performed by others), a higher dimensional analogue of the equidistribution of points on spheres is possible.