2023-05-08 19:18:12 +00:00
|
|
|
\documentclass{article}
|
|
|
|
|
|
|
|
\input{../../preamble}
|
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
\externaldocument[C:1:07:]{Chapter_1_07}[Chapter_1_07.pdf]
|
|
|
|
|
2023-05-11 02:19:18 +00:00
|
|
|
\newcommand{\lean}[1]{\leanref
|
2023-05-11 00:26:01 +00:00
|
|
|
{./Chapter\_1\_11.html\#Apostol.Chapter\_1\_11.#1}
|
|
|
|
{Apostol.Chapter\_1\_11.#1}}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
\begin{document}
|
|
|
|
|
2023-05-10 16:45:42 +00:00
|
|
|
\header{Exercises 1.11}{Tom M. Apostol}
|
|
|
|
|
|
|
|
\section*{Exercise 4}%
|
2023-05-11 02:27:46 +00:00
|
|
|
\label{sec:exercise-4}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
Prove that the greatest-integer function has the properties indicated:
|
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
\subsection*{\verified{Exercise 4a}}%
|
2023-05-11 02:27:46 +00:00
|
|
|
\label{sub:exercise-4a}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
$\floor{x + n} = \floor{x} + n$ for every integer $n$.
|
|
|
|
|
|
|
|
\begin{proof}
|
|
|
|
|
2023-05-11 00:26:01 +00:00
|
|
|
\lean{exercise\_4a}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
\divider
|
|
|
|
|
|
|
|
Let $x$ be a real number and $n$ an integer.
|
|
|
|
Let $m = \floor{x + n}$.
|
|
|
|
By definition of the floor function, $m$ is the unique integer such that
|
|
|
|
$m \leq x + n < m + 1$.
|
|
|
|
Then $m - n \leq x < (m - n) + 1$.
|
|
|
|
That is, $m - n = \floor{x}$.
|
|
|
|
Rearranging terms we see that $m = \floor{x} + n$ as expected.
|
|
|
|
|
2023-05-08 19:18:12 +00:00
|
|
|
\end{proof}
|
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
\subsection*{\verified{Exercise 4b}}%
|
2023-05-11 02:27:46 +00:00
|
|
|
\label{sub:exercise-4b}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
$\floor{-x} =
|
|
|
|
\begin{cases}
|
|
|
|
-\floor{x} & \text{if } x \text{ is an integer}, \\
|
|
|
|
-\floor{x} - 1 & \text{otherwise}.
|
|
|
|
\end{cases}$
|
|
|
|
|
|
|
|
\begin{proof}
|
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
\ \vspace{6pt}
|
|
|
|
|
|
|
|
\lean{exercise\_4b\_1}
|
|
|
|
|
|
|
|
\lean{exercise\_4b\_2}
|
|
|
|
|
|
|
|
\divider
|
|
|
|
|
|
|
|
There are two cases to consider:
|
|
|
|
|
|
|
|
\paragraph{Case 1}%
|
|
|
|
|
|
|
|
Suppose $x$ is an integer.
|
|
|
|
Then $x = \floor{x}$ and $-x = \floor{-x}$.
|
|
|
|
It immediately follows that $$\floor{-x} = -x = -\floor{x}.$$
|
2023-05-09 02:30:59 +00:00
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
\paragraph{Case 2}%
|
|
|
|
|
|
|
|
Suppose $x$ is not an integer.
|
|
|
|
Let $m = \floor{-x}$.
|
|
|
|
By definition of the floor function, $m$ is the unique integer such that
|
|
|
|
$m \leq -x < m + 1$.
|
|
|
|
Equivalently, $-m - 1 < x \leq -m$.
|
|
|
|
Since $x$ is not an integer, it follows $-m - 1 \leq x < -m$.
|
|
|
|
Then, by definition of the floor function, $\floor{x} = -m - 1$.
|
|
|
|
Solving for $m$ yields $$\floor{-x} = m = -\floor{x} - 1.$$
|
|
|
|
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
|
|
|
|
The above two cases are exhaustive. Thus
|
|
|
|
$$\floor{-x} =
|
|
|
|
\begin{cases}
|
|
|
|
-\floor{x} & \text{if } x \text{ is an integer}, \\
|
|
|
|
-\floor{x} - 1 & \text{otherwise}.
|
|
|
|
\end{cases}$$
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
\end{proof}
|
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
\subsection*{\verified{Exercise 4c}}%
|
2023-05-11 02:27:46 +00:00
|
|
|
\label{sub:exercise-4c}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
$\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
|
|
|
|
|
|
|
|
\begin{proof}
|
|
|
|
|
2023-05-11 00:26:01 +00:00
|
|
|
\lean{exercise\_4c}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
\divider
|
|
|
|
|
|
|
|
Rewrite $x$ and $y$ as the sum of their floor and fractional components:
|
|
|
|
$x = \floor{x} + \{x\}$ and $y = \floor{y} + \{y\}$.
|
|
|
|
Now
|
|
|
|
\begin{align}
|
|
|
|
\floor{x + y}
|
|
|
|
& = \floor{\floor{x} + \{x\} + \floor{y} + \{y\}} \nonumber \\
|
|
|
|
& = \floor{\floor{x} + \floor{y} + \{x\} + \{y\}} \nonumber \\
|
|
|
|
& = \floor{x} + \floor{y} + \floor{\{x\} + \{y\}}
|
|
|
|
& \text{\nameref{sub:exercise-4a}} \label{sub:exercise-4c-eq1}
|
|
|
|
\end{align}
|
|
|
|
There are two cases to consider:
|
|
|
|
|
|
|
|
\paragraph{Case 1}%
|
|
|
|
|
|
|
|
Suppose $\{x\} + \{y\} < 1$.
|
|
|
|
Then $\floor{\{x\} + \{y\}} = 0$.
|
|
|
|
Substituting this value into \eqref{sub:exercise-4c-eq1} yields
|
|
|
|
$$\floor{x + y} = \floor{x} + \floor{y}.$$
|
|
|
|
|
|
|
|
\paragraph{Case 2}%
|
|
|
|
|
|
|
|
Suppose $\{x\} + \{y\} \geq 1$.
|
|
|
|
Because $\{x\}$ and $\{y\}$ are both less than $1$, $\{x\} + \{y\} < 2$.
|
|
|
|
Thus $\floor{\{x\} + \{y\}} = 1$.
|
|
|
|
Substituting this value into \eqref{sub:exercise-4c-eq1} yields
|
|
|
|
$$\floor{x + y} = \floor{x} + \floor{y} + 1.$$
|
|
|
|
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
|
|
|
|
Since the above two cases are exhaustive, it follows
|
|
|
|
$\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
|
|
|
|
|
2023-05-08 19:18:12 +00:00
|
|
|
\end{proof}
|
|
|
|
|
2023-05-10 16:45:42 +00:00
|
|
|
\subsection*{\proceeding{Exercise 4d}}%
|
2023-05-11 02:27:46 +00:00
|
|
|
\label{sub:exercise-4d}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
$\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
|
|
|
|
|
|
|
|
\begin{proof}
|
|
|
|
|
2023-05-11 00:26:01 +00:00
|
|
|
\lean{exercise\_4d}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
\divider
|
|
|
|
|
|
|
|
This is immediately proven by applying Hermite's Identity as shown in
|
|
|
|
\nameref{sec:exercise-5}.
|
|
|
|
|
2023-05-08 19:18:12 +00:00
|
|
|
\end{proof}
|
|
|
|
|
2023-05-10 16:45:42 +00:00
|
|
|
\subsection*{\proceeding{Exercise 4e}}%
|
2023-05-11 02:27:46 +00:00
|
|
|
\label{sub:exercise-4e}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
$\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
|
|
|
|
|
|
|
|
\begin{proof}
|
|
|
|
|
2023-05-11 00:26:01 +00:00
|
|
|
\lean{exercise\_4e}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
\divider
|
|
|
|
|
|
|
|
This is immediately proven by applying Hermite's Identity as shown in
|
|
|
|
\nameref{sec:exercise-5}.
|
|
|
|
|
2023-05-08 19:18:12 +00:00
|
|
|
\end{proof}
|
|
|
|
|
2023-05-10 16:45:42 +00:00
|
|
|
\section*{\proceeding{Exercise 5}}%
|
2023-05-11 02:27:46 +00:00
|
|
|
\label{sec:exercise-5}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
The formulas in Exercises 4(d) and 4(e) suggest a generalization for
|
|
|
|
$\floor{nx}$.
|
|
|
|
State and prove such a generalization.
|
|
|
|
|
2023-05-10 16:45:42 +00:00
|
|
|
\note{The stated generalization is known as "Hermite's Identity."}
|
2023-05-09 17:15:49 +00:00
|
|
|
|
2023-05-08 19:18:12 +00:00
|
|
|
\begin{proof}
|
|
|
|
|
2023-05-11 00:26:01 +00:00
|
|
|
\lean{exercise\_5}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
2023-05-10 16:45:42 +00:00
|
|
|
\divider
|
2023-05-09 17:15:49 +00:00
|
|
|
|
|
|
|
We prove that for all natural numbers $n$ and real numbers $x$, the following
|
|
|
|
identity holds:
|
|
|
|
\begin{equation}
|
|
|
|
\label{sec:exercise-5-eq1}
|
|
|
|
\floor{nx} = \sum_{i=0}^{n-1} \floor{x + \frac{i}{n}}
|
|
|
|
\end{equation}
|
|
|
|
By definition of the floor function, $x = \floor{x} + r$ for some
|
|
|
|
$r \in \ico{0}{1}$.
|
|
|
|
Define $S$ as the partition of non-overlapping subintervals
|
|
|
|
$$\ico{0}{\frac{1}{n}}, \ico{\frac{1}{n}}{\frac{2}{n}}, \ldots,
|
|
|
|
\ico{\frac{n-1}{n}}{1}.$$
|
|
|
|
By construction, $\cup\; S = \ico{0}{1}$.
|
|
|
|
Therefore there exists some $j \in \mathbb{N}$ such that
|
|
|
|
\begin{equation}
|
|
|
|
\label{sec:exercise-5-eq2}
|
|
|
|
r \in \ico{\frac{j}{n}}{\frac{j+1}{n}}.
|
|
|
|
\end{equation}
|
|
|
|
With these definitions established, we now show the left- and right-hand sides
|
|
|
|
of \eqref{sec:exercise-5-eq1} evaluate to the same number.
|
|
|
|
|
|
|
|
\paragraph{Left-Hand Side}%
|
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
Consider the left-hand side of identity \eqref{sec:exercise-5-eq1}.
|
2023-05-09 17:15:49 +00:00
|
|
|
By \eqref{sec:exercise-5-eq2}, $nr \in \ico{j}{j + 1}$.
|
|
|
|
Therefore $\floor{nr} = j$.
|
|
|
|
Thus
|
2023-05-09 22:28:45 +00:00
|
|
|
\begin{align}
|
2023-05-09 17:15:49 +00:00
|
|
|
\floor{nx}
|
|
|
|
& = \floor{n(\floor{x} + r)} \nonumber \\
|
|
|
|
& = \floor{n\floor{x} + nr} \nonumber \\
|
|
|
|
& = \floor{n\floor{x}} + \floor{nr}. \nonumber
|
2023-05-10 16:45:42 +00:00
|
|
|
& \text{\nameref{sub:exercise-4a}} \\
|
2023-05-09 17:15:49 +00:00
|
|
|
& = \floor{n\floor{x}} + j \nonumber \\
|
2023-05-09 22:28:45 +00:00
|
|
|
& = n\floor{x} + j. \label{sec:exercise-5-eq3}
|
|
|
|
\end{align}
|
2023-05-09 17:15:49 +00:00
|
|
|
|
|
|
|
\paragraph{Right-Hand Side}%
|
|
|
|
|
|
|
|
Now consider the right-hand side of identity \eqref{sec:exercise-5-eq1}.
|
|
|
|
We note each summand, by construction, is the floor of $x$ added to a
|
|
|
|
nonnegative number less than one.
|
|
|
|
Therefore each summand contributes either $\floor{x}$ or $\floor{x} + 1$ to
|
|
|
|
the total.
|
|
|
|
Letting $z$ denote the number of summands that contribute $\floor{x} + 1$,
|
|
|
|
we have
|
|
|
|
\begin{equation}
|
|
|
|
\label{sec:exercise-5-eq4}
|
|
|
|
\sum_{i=0}^{n-1} \floor{x + \frac{i}{n}} = n\floor{x} + z.
|
|
|
|
\end{equation}
|
|
|
|
The value of $z$ corresponds to the number of indices $i$ that satisfy
|
|
|
|
$$\frac{i}{n} \geq 1 - r.$$
|
|
|
|
By \eqref{sec:exercise-5-eq2}, it follows
|
|
|
|
\begin{align*}
|
|
|
|
1 - r
|
|
|
|
& \in \ioc{1 - \frac{j+1}{n}}{1-\frac{j}{n}} \\
|
|
|
|
& = \ioc{\frac{n - j - 1}{n}}{\frac{n - j}{n}}.
|
|
|
|
\end{align*}
|
|
|
|
Thus we can determine the value of $z$ by instead counting the number of
|
|
|
|
indices $i$ that satisfy $$\frac{i}{n} \geq \frac{n - j}{n}.$$
|
|
|
|
Rearranging terms, we see that $i \geq n - j$ holds for
|
|
|
|
$z = (n - 1) - (n - j) + 1 = j$ of the $n$ summands.
|
|
|
|
Substituting the value of $z$ into \eqref{sec:exercise-5-eq4} yields
|
|
|
|
\begin{equation}
|
|
|
|
\label{sec:exercise-5-eq5}
|
|
|
|
\sum_{i=0}^{n-1} \floor{x + \frac{i}{n}} = n\floor{x} + j.
|
|
|
|
\end{equation}
|
|
|
|
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
|
|
|
|
Since \eqref{sec:exercise-5-eq3} and \eqref{sec:exercise-5-eq5} agree with
|
|
|
|
one another, it follows identity \eqref{sec:exercise-5-eq1} holds.
|
|
|
|
|
2023-05-08 19:18:12 +00:00
|
|
|
\end{proof}
|
|
|
|
|
2023-05-10 16:45:42 +00:00
|
|
|
\section*{\unverified{Exercise 6}}%
|
2023-05-11 02:27:46 +00:00
|
|
|
\label{sec:exercise-6}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are
|
|
|
|
integers.
|
|
|
|
Let $f$ be a nonnegative function whose domain is the interval $[a, b]$, where
|
|
|
|
$a$ and $b$ are integers, $a < b$.
|
|
|
|
Let $S$ denote the set of points $(x, y)$ satisfying $a \leq x \leq b$,
|
|
|
|
$0 < y \leq f(x)$.
|
|
|
|
Prove that the number of lattice points in $S$ is equal to the sum
|
|
|
|
$$\sum_{n=a}^b \floor{f(n)}.$$
|
|
|
|
|
|
|
|
\begin{proof}
|
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
Let $i = a, \ldots, b$ and define $S_i = \mathbb{N} \cap \ioc{0}{f(i)}$.
|
|
|
|
By construction, the number of lattice points in $S$ is
|
|
|
|
\begin{equation}
|
|
|
|
\label{sec:exercise-6-eq1}
|
|
|
|
\sum_{n = a}^b \abs{S_n}.
|
|
|
|
\end{equation}
|
|
|
|
All that remains is to show $\abs{S_i} = \floor{f(i)}$.
|
|
|
|
There are two cases to consider:
|
|
|
|
|
|
|
|
\paragraph{Case 1}%
|
|
|
|
|
|
|
|
Suppose $f(i)$ is an integer.
|
|
|
|
Then the number of integers in $\ioc{0}{f(i)}$ is $f(i) = \floor{f(i)}$.
|
|
|
|
|
|
|
|
\paragraph{Case 2}%
|
|
|
|
|
|
|
|
Suppose $f(i)$ is not an integer.
|
|
|
|
Then the number of integers in $\ioc{0}{f(i)}$ is the same as that of
|
|
|
|
$\ioc{0}{\floor{f(i)}}$.
|
|
|
|
Once again, that number is $\floor{f(i)}$.
|
|
|
|
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
|
|
|
|
By cases 1 and 2, $\abs{S_i} = \floor{f(i)}$.
|
|
|
|
Substituting this identity into \eqref{sec:exercise-6-eq1} finishes the
|
|
|
|
proof.
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
\end{proof}
|
|
|
|
|
2023-05-10 16:45:42 +00:00
|
|
|
\section*{Exercise 7}%
|
2023-05-11 02:27:46 +00:00
|
|
|
\label{sec:exercise-7}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
If $a$ and $b$ are positive integers with no common factor, we have the formula
|
|
|
|
$$\sum_{n=1}^{b-1} \floor{\frac{na}{b}} = \frac{(a - 1)(b - 1)}{2}.$$
|
|
|
|
When $b = 1$, the sum on the left is understood to be $0$.
|
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
\note{When $b = 1$, the proofs of (a) and (b) are trivial. We continue under the
|
|
|
|
assumption $b > 1$.}
|
|
|
|
|
2023-05-10 16:45:42 +00:00
|
|
|
\subsection*{\unverified{Exercise 7a}}%
|
2023-05-11 02:27:46 +00:00
|
|
|
\label{sub:exercise-7a}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
Derive this result by a geometric argument, counting lattice points in a right
|
|
|
|
triangle.
|
|
|
|
|
|
|
|
\begin{proof}
|
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
Let $f \colon [1, b - 1] \rightarrow \mathbb{R}$ be given by $f(x) = ax / b$.
|
|
|
|
Let $S$ denote the set of points $(x, y)$ satisfying $1 \leq x \leq b - 1$,
|
|
|
|
$0 < y \leq f(x)$.
|
|
|
|
By \nameref{sec:exercise-6}, the number of lattice points of $S$ is equal to
|
|
|
|
the sum
|
|
|
|
\begin{equation}
|
|
|
|
\label{sub:exercise-7a-eq1}
|
|
|
|
\sum_{n=1}^{b-1} \floor{f(n)} = \sum_{n=1}^{b-1} \floor{\frac{na}{b}}.
|
|
|
|
\end{equation}
|
|
|
|
Define $T$ to be the triangle of width $w = b$ and height $h = f(b) = a$
|
|
|
|
as $$T = \{ (x, y) : 0 < x < b, 0 < y \leq f(x) \}.$$
|
|
|
|
By construction, $T$ does not introduce any additional lattice points.
|
|
|
|
Thus $S$ and $T$ have the same number of lattice points.
|
|
|
|
Let $H_L$ denote the number of boundary points on $T$'s hypotenuse.
|
|
|
|
We prove that (i) $H_L = 2$ and (ii) that $T$ has $\frac{(a - 1)(b - 1)}{2}$
|
|
|
|
lattice points.
|
|
|
|
|
|
|
|
\paragraph{(i)}%
|
|
|
|
\label{par:exercise-7a-i}
|
|
|
|
|
|
|
|
Consider the line $L$ overlapping the hypotenuse of $T$.
|
|
|
|
By construction, $T$'s hypotenuse has endpoints $(0, 0)$ and $(b, a)$.
|
|
|
|
By hypothesis, $a$ and $b$ are positive, excluding the possibility of $L$
|
|
|
|
being vertical.
|
|
|
|
Define the slope of $L$ as $$m = \frac{a}{b}.$$
|
|
|
|
$H_L$ coincides with the number of indices $i = 0, \ldots, b$ such that
|
|
|
|
$(i, i * m)$ is a lattice point.
|
|
|
|
But $a$ and $b$ are coprime by hypothesis and $i \leq b$.
|
|
|
|
Thus $i * m$ is an integer if and only if $i = 0$ or $i = b$.
|
|
|
|
Thus $H_L = 2$.
|
|
|
|
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
|
|
|
|
Next we count the number of lattice points in $T$.
|
|
|
|
Let $R$ be the overlapping retangle of width $w$ and height $h$, situated
|
|
|
|
with bottom-left corner at $(0, 0)$.
|
|
|
|
Let $I_R$ denote the number of interior lattice points of $R$.
|
|
|
|
Let $I_T$ and $B_T$ denote the interior and boundary lattice points of $T$
|
|
|
|
respectively.
|
|
|
|
By \nameref{C:1:07:sub:exercise-4b-eq2},
|
|
|
|
\begin{align}
|
|
|
|
I_T
|
|
|
|
& = \frac{1}{2}(I_R - (H_L - 2)) \nonumber \\
|
|
|
|
& = \frac{1}{2}(I_R - (2 - 2))
|
|
|
|
& \text{\nameref{par:exercise-7a-i}} \nonumber \\
|
|
|
|
& = \frac{1}{2}I_R. & \label{sub:exercise-7a-eq2}
|
|
|
|
\end{align}
|
|
|
|
Furthermore, since both the adjacent and opposite side of $T$ are not
|
|
|
|
included in $T$ and there exist no lattice points on $T$'s hypotenuse
|
|
|
|
besides the endpoints, it follows
|
|
|
|
\begin{equation}
|
|
|
|
\label{sub:exercise-7a-eq3}
|
|
|
|
B_T = 0.
|
|
|
|
\end{equation}
|
|
|
|
Thus the number of lattice points of $T$ equals
|
|
|
|
\begin{align}
|
|
|
|
I_T + B_T
|
|
|
|
& = I_T & \eqref{sub:exercise-7a-eq3} \nonumber \\
|
|
|
|
& = \frac{1}{2}I_R & \eqref{sub:exercise-7a-eq2} \nonumber \\
|
|
|
|
& = \frac{(b - 1)(a - 1)}{2}.
|
|
|
|
& \text{\nameref{C:1:07:sub:exercise-4a}}
|
|
|
|
\label{sub:exercise-7a-eq4}
|
|
|
|
\end{align}
|
|
|
|
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
|
|
|
|
By \eqref{sub:exercise-7a-eq1} the number of lattice points of $S$ is equal
|
|
|
|
to the sum $$\sum_{n=1}^{b-1} \floor{\frac{na}{b}}.$$
|
|
|
|
But the number of lattice points of $S$ is the same as that of $T$.
|
|
|
|
By \eqref{sub:exercise-7a-eq4}, the number of lattice points in $T$ is equal
|
|
|
|
to $$\frac{(b - 1)(a - 1)}{2}.$$
|
|
|
|
Thus $$\sum_{n=1}^{b-1} \floor{\frac{na}{b}} = \frac{(a - 1)(b - 1)}{2}.$$
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
\end{proof}
|
|
|
|
|
2023-05-10 16:45:42 +00:00
|
|
|
\subsection*{\proceeding{Exercise 7b}}%
|
2023-05-11 02:27:46 +00:00
|
|
|
\label{sub:exercise-7b}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
Derive the result analytically as follows:
|
|
|
|
By changing the index of summation, note that
|
|
|
|
$\sum_{n=1}^{b-1} \floor{na / b} = \sum_{n=1}^{b-1} \floor{a(b - n) / b}$.
|
|
|
|
Now apply Exercises 4(a) and (b) to the bracket on the right.
|
|
|
|
|
|
|
|
\begin{proof}
|
|
|
|
|
2023-05-11 00:26:01 +00:00
|
|
|
\lean{exercise\_7b}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
2023-05-11 19:35:05 +00:00
|
|
|
\divider
|
|
|
|
|
|
|
|
Let $n = 1, \ldots, b - 1$.
|
|
|
|
By hypothesis, $a$ and $b$ are coprime.
|
|
|
|
Furthermore, $n < b$ for all values of $n$.
|
|
|
|
Thus $an / b$ is not an integer.
|
|
|
|
By \nameref{sub:exercise-4b},
|
|
|
|
\begin{equation}
|
|
|
|
\label{sub:exercise-7b-eq1}
|
|
|
|
\floor{-\frac{an}{b}} = -\floor{\frac{an}{b}} - 1.
|
|
|
|
\end{equation}
|
|
|
|
Consider the following:
|
|
|
|
\begin{align*}
|
|
|
|
\sum_{n=1}^{b-1} \floor{\frac{na}{b}}
|
|
|
|
& = \sum_{n=1}^{b-1} \floor{\frac{a(b - n)}{b}} \\
|
|
|
|
& = \sum_{n=1}^{b-1} \floor{\frac{ab - an}{b}} \\
|
|
|
|
& = \sum_{n=1}^{b-1} \floor{-\frac{an}{b} + a} \\
|
|
|
|
& = \sum_{n=1}^{b-1} \floor{-\frac{an}{b}} + a.
|
|
|
|
& \text{\nameref{sub:exercise-4a}} \\
|
|
|
|
& = \sum_{n=1}^{b-1} -\floor{\frac{an}{b}} - 1 + a
|
|
|
|
& \eqref{sub:exercise-7b-eq1} \\
|
|
|
|
& = -\sum_{n=1}^{b-1} \floor{\frac{an}{b}} - \sum_{n=1}^{b-1} 1 +
|
|
|
|
\sum_{n=1}^{b-1} a \\
|
|
|
|
& = -\sum_{n=1}^{b-1} \floor{\frac{an}{b}} - (b - 1) + a(b - 1).
|
|
|
|
\end{align*}
|
|
|
|
Rearranging the above yields
|
|
|
|
$$2\sum_{n=1}^{b-1} \floor{\frac{an}{b}} = (a - 1)(b - 1).$$
|
|
|
|
Dividing both sides of the above identity concludes the proof.
|
|
|
|
|
2023-05-08 19:18:12 +00:00
|
|
|
\end{proof}
|
|
|
|
|
2023-05-12 13:05:31 +00:00
|
|
|
\section*{\proceeding{Exercise 8}}%
|
2023-05-11 02:27:46 +00:00
|
|
|
\label{sec:exercise-8}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
Let $S$ be a set of points on the real line.
|
|
|
|
The \textit{characteristic function} of $S$ is, by definition, the function
|
2023-05-12 13:05:31 +00:00
|
|
|
$\mathcal{X}_S$ such that $\mathcal{X}_S(x) = 1$ for every $x$ in $S$, and
|
|
|
|
$\mathcal{X}_S(x) = 0$ for those $x$ not in $S$.
|
2023-05-08 19:18:12 +00:00
|
|
|
Let $f$ be a step function which takes the constant value $c_k$ on the $k$th
|
|
|
|
open subinterval $I_k$ of some partition of an interval $[a, b]$.
|
|
|
|
Prove that for each $x$ in the union $I_1 \cup I_2 \cup \cdots \cup I_n$ we have
|
2023-05-12 13:05:31 +00:00
|
|
|
$$f(x) = \sum_{k=1}^n c_k\mathcal{X}_{I_k}(x).$$
|
2023-05-08 19:18:12 +00:00
|
|
|
This property is described by saying that every step function is a linear
|
|
|
|
combination of characteristic functions of intervals.
|
|
|
|
|
|
|
|
\begin{proof}
|
|
|
|
|
2023-05-12 13:05:31 +00:00
|
|
|
Let $x \in I_1 \cup I_2 \cup \cdots \cup I_n$ and $N = \{1, \ldots, n\}$.
|
|
|
|
Let $k \in N$ such that $x \in I_k$.
|
|
|
|
Consider an arbitrary $j \in N - \{k\}$.
|
|
|
|
By definition of a partition, $I_j \cap I_k = \emptyset$.
|
|
|
|
That is, $I_j$ and $I_k$ are disjoint for all $j \in N - \{k\}$.
|
|
|
|
Therefore, by definition of the characteristic function,
|
|
|
|
$\mathcal{X}_{I_k}(x) = 1$ and $\mathcal{X}_{I_j}(x) = 0$ for all
|
|
|
|
$j \in N - \{k\}$.
|
|
|
|
Thus
|
|
|
|
\begin{align*}
|
|
|
|
f(x)
|
|
|
|
& = c_k \\
|
|
|
|
& = (c_k)(1) + \sum\nolimits_{j \in N - \{k\}} (c_j)(0) \\
|
|
|
|
& = c_k\mathcal{X}_{I_k}(x) +
|
|
|
|
\sum\nolimits_{j \in N - \{k\}} c_j\mathcal{X}_{I_j}(x) \\
|
|
|
|
& = \sum_{k=1}^n c_k\mathcal{X}_{I_k}(x).
|
|
|
|
\end{align*}
|
2023-05-08 19:18:12 +00:00
|
|
|
|
|
|
|
\end{proof}
|
|
|
|
|
|
|
|
\end{document}
|