In this thesis we compute an explicit Plancherel fromula for PGL_2(F) where F is a non-archimedean local field. Let G be connected reductive group over a non-archimedean local field F. We show that we can obtain types and covers as defined by Kutzko and Bushnell for G/Z coming from types and covers of G in a very explicit way. We then compute those types and covers for GL_2(F ) which give rise to all types and covers for PGL_2(F) that are in the principal series. The Hecke algebra is a Hilbert algebra and has a measure associated to it called Plancherel measure of the Hecke algebra. We have that computing the Plancherel measure for PGL_2(F) essentially reduces to computing the Plancherel measure for the Hecke algebra for every type. We get that the Hacke algebras come in two flavors; they are either the group ring of the integers or they are a free algebra in two generators s_1, s_2 subject to the relations s_1^2=1 and s_2^2=(q^{-1/2}-q^{-1/2})s_2+1, where q is the order of the residue field. The Plancherel measure for both algebras are known, as a result we obtain the Plancherel measure for PGL_2(F).

Let S and T be numerical semigroups and let k be a positive integer. We say that S is the quotient of T by k if an integer x belongs to S if and only if kx belongs to T. Given any integer k larger than 1 (resp., larger than 2), every numerical semigroup S is the quotient T/k of infinitely many symmetric (resp., pseudo-symmetric) numerical semigroups T by k. Related examples, probabilistic results, and applications to ring theory are shown. Given an arbitrary positive integer k, it is not true in general that every numerical semigroup S is the quotient of infinitely many numerical semigroups of maximal embedding dimension by k. In fact, a numerical semigroup S is the quotient of infinitely many numerical semigroups of maximal embedding dimension by each positive integer k larger than 1 if and only if S is itself of maximal embedding dimension. Nevertheless, for each numerical semigroup S, for all sufficiently large positive integers k, S is the quotient of a numerical semigroup of maximal embedding dimension by k. Related results and examples are also given.

In this report we give an overview of some of the major results concerning the multiplicities of linear recurrence sequences. We first investigate binary recurrence sequences where we exhibit a result due to Beukers and a result due to Brindza, Pint&eacute;r and Schmidt. We then investigate ternary recurrences and exhibit a result due to Beukers building on work of Beukers and Tijdeman. The last two chapters deal with a very important result due to Schmidt in which we bound the zero-multiplicity of a linear recurrence sequence of order <em>t</em> by a function involving <em>t</em> alone. Moreover we improve on Schmidt's bound by making some minor changes to his argument.

We consider the greatest prime factors of integers of certain form.

We answer to a Wintner's question concerning the sequence of integers composed of primes from a given set. The results generalize and develop the answer to Wintner’s question due to Tijdeman.

In this report we give an overview of some of the major results concerning the multiplicities of linear recurrence sequences. We first investigate binary recurrence sequences where we exhibit a result due to Beukers and a result due to Brindza, Pint&eacute;r and Schmidt. We then investigate ternary recurrences and exhibit a result due to Beukers building on work of Beukers and Tijdeman. The last two chapters deal with a very important result due to Schmidt in which we bound the zero-multiplicity of a linear recurrence sequence of order <em>t</em> by a function involving <em>t</em> alone. Moreover we improve on Schmidt's bound by making some minor changes to his argument.

We consider the greatest prime factors of integers of certain form.

We answer to a Wintner's question concerning the sequence of integers composed of primes from a given set. The results generalize and develop the answer to Wintner’s question due to Tijdeman.

For any finite set B and a subset A⊆B, we define the density of A in B to be the value α=|A|/|B|. Roth's famous theorem, proven in 1953, states that there is a constant C>0, such that if A⊆{1,...,N} for a positive integer N and A has density α in {1,...,N} with α>C/loglog N, then A contains a non-trivial arithmetic progression of length three (3AP). The proof of this relies on the following dichotomy: either 1) A looks like a random set and the number of 3APs in A is close to the probabilistic expected value, or 2) A is more structured and consequently, there is a progression P of about length α√N on which A∩P has α(1+cα) for some c>0. If 1) occurs, then we are done. If 2) occurs, then we identify P with {1,...,|P|} and repeat the above argument, whereby the density increases at each iteration of the dichotomy. Due to the density increase in case 2), an argument of this type is called a density increment argument. The density increment is obtained by studying the Fourier transforms of the characterstic function of A and extracting a structure out of A. Improving the lower bound for α is still an active area of research and all improvements so far employ a density increment. Two of the most recent results are α>C(loglog N/log N)^{1/2} by Bourgain in 1999 and α>C(loglog N)^5/log N by Sanders in 2010. This thesis is a survey of progresses in Roth's theorem, with a focus on these last two results. Attention was given to unifying the language in which the results are discussed and simplifying the presentation.

The Erdős discrepancy problem asks, "Does there exist a sequence t = {t_i}_{1≤i<∞} with each t_i ∈ {-1,1} and a constant c such that |∑_{1≤i≤n} t_{id}| ≤ c for all n,c ∈ ℕ = {1,2,3,...}?" The discrepancy of t equals sup_{n≥1} |∑_{1≤i≤n} t_{id}|. Erdős conjectured in 1957 that no such sequence exists. We examine versions of this problem with fixed values for c and where the values of d are restricted to particular subsets of ℕ. By examining a wide variety of different subsets, we hope to learn more about the original problem. When the values of d are restricted to the set {1,2,4,8,...}, we show that there are exactly two infinite {-1,1} sequences with discrepancy bounded by 1 and an uncountable number of in nite {-1,1} sequences with discrepancy bounded by 2. We also show that the number of {-1,1} sequences of length n with discrepancy bounded by 1 is 2^{s2(n)} where s2(n) is the number of 1s in the binary representation of n. When the values of d are restricted to the set {1,b,b^2,b^3,...} for b > 2, we show there are an uncountable number of infinite sequences with discrepancy bounded by 1. We also give a recurrence for the number of sequences of length n with discrepancy bounded by 1. When the values of d are restricted to the set {1,3,5,7,..} we conjecture that there are exactly 4 in finite sequences with discrepancy bounded by 1 and give some experimental evidence for this conjecture. We give descriptions of the lexicographically least sequences with D-discrepancy c for certain values of D and c as fixed points of morphisms followed by codings. These descriptions demonstrate that these automatic sequences. We introduce the notion of discrepancy-1 maximality and prove that {1,2,4,8,...} and {1,3,5,7,...} are discrepancy-1 maximal while {1,b,b^2,...} is not for b > 2. We conclude with some open questions and directions for future work.