507 – Homework 3

This set is due Monday, November 8.

  1. The goal of this exercise is to use the method of generating functions to verify some results of D.J. Newman and Paul Erdös.

    For the purposes of this exercise, an arithmetic progression is a subset of {{\mathbb N}=\{0,1,\dots\}} of the form {A=\{a+bi\mid i\in{\mathbb N}\}} for some fixed naturals {a,b} with {b\ne0}, and we call {b} the common difference of {A}.

    Recall that {A_1,\dots,A_k} is a partition of a set {S} iff the following hold:

    1. {A_i\ne\emptyset} for all {i}.
    2. {A_i\cap A_j=\emptyset} for all {i\ne j}.
    3. {\bigcup_{i=1}^k A_i=S}.

    We want to show the following:

    Suppose that {{\mathbb N}} is partitioned into finitely many arithmetic progressions {A_1,\dots,A_k}. Then either {k=1}, or at least two of the common differences coincide. Moreover, if {b_1,\dots,b_k} denote the common differences, then

    \displaystyle  1=\frac1{b_1}+\dots+\frac1{b_k}.

    To do this, we can use the method of generating functions as follows: Associate to {A\subseteq{\mathbb N}} the generating function {f_A} of {\chi_A}:

    \displaystyle  f_A(x)=\sum_{a\in A}x^a.

    1. Verify that

      \displaystyle  \sum_{j=1}^k f_{A_j}(x)=f_{{\mathbb N}}(x)=\frac1{1-x},

      where the last equality is valid for all {{}|x|<1}. (We will need that it is valid for all complex numbers {x} with {|x|<1}. You may assume this.)

    2. For {1\le j\le k}, define {a_i} so that {A_j=\{a_j+nb_j\mid n\in{\mathbb N}\}}. Find a formula for {f_{A_j}(x)} similar to the formula above for {f_A}, involving {a_j} and {b_j}.
    3. Suppose that {k>1} and the {b_j} are all different. Let {b} be the largest of them, and consider

      \displaystyle  (1-x^b)\sum_j f_{A_j}=\frac{1-x^b}{1-x}.

      Derive a contradiction by letting {x\rightarrow e^{2\pi i/b}}.

    4. Consider

      \displaystyle  (1-x)\sum_j f_{A_j}=1,

      and use L’Hôpital’s rule to deduce that

      \displaystyle  \sum_j \frac 1{b_j}=1.

  2. For {k} a positive integer and {m\in{\mathbb Z}}, let

    \displaystyle  g(m,k)=\sum_{n=0}^{k-1}e^{2\pi i m n^2/k}.  

    Suppose that {a,b} are positive integers and {(a,b)=1}. Show that for any {t\in{\mathbb Z}},

    \displaystyle  g(tb,a)g(ta,b)=g(t,ab).

    For this, you may want to verify first that the numbers of the form {ar+bs} with {0\le r<b} and {0\le s<a} run through a complete residue system {\pmod {ab}}.

  3. This problem requires some basic facts about matrices, eigenvalues, and traces. A good reference is Sheldon Axler, Linear algebra done right. Springer, 2nd edition (1997), but feel free to ask me if you are not familiar or do not feel comfortable with linear algebra. Let {\zeta=e^{2\pi i/k}} where {k>1} is an odd integer. Consider the {k\times k} matrix {M} with entry {\zeta^{ab}/\sqrt k} in row {a}, column {b} ({a,b=0,\dots,k-1}).
    1. Show that {M^2} is a permutation matrix, i.e., all its entries are 0s or 1s, and there is exactly one 1 in each column and each row.
    2. Show that {M^4=I}, and conclude that the eigenvalues of {M} are all among {1,-1,i,-i}.
    3. Verify that the trace of {M^2} is 1 to conclude that the eigenvalues of {M^2} are 1 with multiplicity {(k+1)/2} and {-1} with multiplicity {(k-1)/2}.
  4. Solve exercises 4.6.9–11 from the textbook.

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