Problems in Combinatorial and Analytic Number Theory

Abstract

We focus on three problems in number theory. The first problem studies the random Fibonacci tree, which is an infinite binary tree with non-negative integers at each node. The root consists of the number 1 with a single child, also the number 1. We define the tree recursively in the following way: if x is the parent of y, then y has two children, namely |x-y| and x+y. This tree was studied by Benoit Rittaud who proved that any pair of integers a,b that are coprime occur as a parent-child pair infinitely often. We extend his results by determining the probability that a random infinite walk in this tree contains exactly one pair (1,1), that being at the root of the tree. Also, we give tight upper and lower bounds on the number of occurrences of any specific coprime pair (a,b) at any given fixed depth in the tree. The second problem studies sieve methods in combinatorics. We apply the Turan sieve and the simple sieve developed by Ram Murty and Yu-Ru Liu to study problems in random graph theory. More specifically, we obtain bounds on the probability of a graph having diameter 2 (or diameter 3 in the case of bipartite graphs). An interesting feature revealed in these results is that the Turan sieve and the simple sieve ``almost completely'' complement each other.\par The third problem studies the Mahler measure of a polynomial with integer coefficients. We give a lower bound of the Mahler measure on a set of polynomials that are ``almost" reciprocal. Here ``almost" reciprocal means that the outermost coefficients of each polynomial mirror each other in proportion, while this pattern breaks down for the innermost coefficients.