Lecture notesHere are some typeset notes from a small selection of courses I took as an undergraduate student in mathematics at the University of Cambridge, ordered by Tripos part. Part IA
Part IBPart IIPart IIIHere are two further comments regarding Chapter 3 on Quadratic Recurrence, which probably should have made it into the Appendix. In Chapter 3, we eventually prove that there exists a positive absolute constant C such that, for every positive epsilon and every real number alpha, there exists an integer d in [1,C epsilon**-5 + 1] such that |e(alpha d**2) - 1| ≤ epsilon. Given epsilon, on page 13 the proof begins by implicitly assuming that alpha is a rational of the form a/N, where we may also assume N to be prime, because the set of such rationals is dense in the set of real numbers. On page 14, without further explanation we assume that the set A = {a, 4a, 9a, ..., k**2 a} in Z/NZ has cardinality k. Evidently, this need not be true for all k and N. We note that, since from alpha = a/N we may assume that a and N are coprime, we can consider the cardinality of the set B = {i**2 : 1 ≤ i ≤ k} in Z/NZ. Recall that we may assume that N is prime. If now 1 ≤ k < N then as Z/NZ is a field we observe that the multiset B covers every element of (Z/NZ)^{×} at most twice. Hence A has cardinality at least k/2. Barring the extra factor of 2, this allows the proof to go through exactly as stated, only changing the absolute constant C to C'. We are done by observing that, because we ultimately obtain the sufficient condition k > C' epsilon**-5, we need only choose N sufficiently large, e.g. N > C' epsilon**-5 + 1 will do. Again, we note that the set of all rational numbers of the form a/N with N > C' epsilon**-5 + 1 prime and coprime to a is dense in the set of real numbers. |
28 July 2008