## On strong measure zero sets

December 6, 2013

I meant to write a longer blog entry on strong measure zero sets (on the real line $\mathbb R$), but it is getting too long, so it may take me more than I expected. For now, let me record here an argument showing the following:

Theorem. If $X$ is a strong measure zero set and $F$ is a closed measure zero set, then $X+F$ has measure zero.

The argument is similar to the one in

Janusz Pawlikowski. A characterization of strong measure zero sets, Israel J. Math., 93 (1), (1996), 171-183. MR1380640 (97f:28003),

where the result is shown for strong measure zero subsets of $\{0,1\}^{\mathbb N}$. This is actually the easy direction of Pawlikowski’s result, showing that this condition actually characterizes strong measure zero sets, that is, if $X+F$ is measure zero for all closed measure zero sets $F$, then $X$ is strong measure zero. (Since this was intended for my analysis course, and I do not see how to prove Pawlikowski’s argument without some appeal to results in measure theory, I am only including here the easy direction.) Pawlikowski’s argument actually generalizes an earlier key result of Galvin, Mycielski, and Solovay, who proved that a set $X$ has strong measure zero iff it can be made disjoint from any given meager set by translation, that is, iff for any $G$ meager there is a real $r$ with $X+r$ disjoint from $G$.

I proceed with the (short) proof after the fold.

## The day is coming

November 19, 2013

(BSU, last year. See also here and here.)

## Riemann on Riemann sums

November 16, 2013

Though Riemann sums had been considered earlier, at least in particular cases (for example, by Cauchy), the general version we consider today was introduced by Riemann, when studying problems related to trigonometric series, in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. This was his Habilitationsschrift, from 1854, published posthumously in 1868.

Riemann’s papers (in German) have been made available by the Electronic Library of Mathematics, see here. The text in question appears in section 4, Ueber den Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit. The translation below is as in

• A source book in classical analysis. Edited by Garrett Birkhoff. With the assistance of Uta Merzbach. Harvard University Press, Cambridge, Mass., 1973. MR0469612 (57 #9395).

Also zuerst: Was hat man unter $\displaystyle \int_a^b f(x) \, dx$ zu verstehen?

Um dieses festzusetzen, nehmen wir zwischen $a$ und $b$ der Grösse nach auf einander folgend, eine Reihe von Werthen $x_1, x_2,\ldots, x_{n-1}$ an und bezeichnen der Kürze wegen $x_1 - a$ durch $\delta_1$, $x_2 - x_1$ durch $\delta_2,\ldots,$ $b - x_{n-1}$ durch $\delta_n$ und durch $\varepsilon$ einen positiven ächten Bruch.  Es wird alsdann der Werth der Summe

$\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots$ $\displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)$

von der Wahl der Intervalle $\delta$ und der Grössen $\varepsilon$ abhängen.  Hat sie nun die Eigenschaft, wie auch $\delta$ und $\varepsilon$ gewählt werden mögen, sich einer festen Grenze $A$ unendlich zu nähern, sobald sämmtliche $\delta$ unendlich klein werden, so heisst dieser Werth $\displaystyle \int_a^b f(x) \, dx$.

In Birkhoff’s book:

First of all: What is to be understood by $\displaystyle \int_a^b f(x)\,dx$?

In order to establish this, we take the sequence of values $x_1,x_2,\ldots, x_{n-1}$ lying between $a$ and $b$ and ordered by size, and, for brevity, denote $x_1 - a$ by $\delta_1$, $x_2 - x_1$ by $\delta_2,\ldots,$ $b - x_{n-1}$ by $\delta_n$, and proper positive fractions by $\varepsilon_i$. Then the value of the sum

$\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots$ $\displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)$

will depend on the choice of the intervals $\delta_i$ and the quantities $\varepsilon_i$. If it has the property that, however the $\delta_i$ and the $\varepsilon_i$ may be chosen, it tends to a fixed limit $A$ as soon as all the $\delta_i$ become infinitely small, then this value is called $\displaystyle \int_a^b f(x) \, dx$.

(Of  course, in modern presentations, we use $\Delta_i$ instead of $\delta_i$, and say that the $\delta_i$ approach $0$ rather than become infinitely small. In fact, we tend to call the collection of data $x_1,\dots,x_{n-1}$, $\varepsilon_1,\dots,\varepsilon_n$ a tagged partition of ${}[a,b]$, and call the maximum of the $x_{i+1}-x_i$ the mesh or norm of the partition.)

## Analysis – HW 5 – Newton’s method

November 16, 2013

This set is due Friday, December 6, at the beginning of lecture.

Newton’s method was introduced by Newton on De analysi in 1669. It was originally restricted to polynomials; his example in Methodus fluxionum was the cubic equation

$x^3-2x-5=0.$

Raphson simplified its description in 1690. The modern presentation, in full generality, is due to Simpson in 1740. Here, we are mostly interested in the dynamics of Newton’s method on polynomials.

## AlgoRythmics

November 15, 2013

This link should take you to the YouTube channel of Algo-rythmics, or see their website.

Different sorting algorithms (bubble sort, insertion sort, quicksort, selection sort, shell sort) illustrated through folk dance.

## Shirt

November 8, 2013

Isn’t this cute?

All the kids in Francisco’s class drew themselves. And then they got a class shirt with all the drawings. I think the idea is to wear it every Friday.

I got one for myself as well.

(Bemmy is the one farthest to the right, in the front row.)

## Weierstrass function

November 7, 2013

Weierstrass function from 1872 is the function $f=f_{a,b}$ defined by

$\displaystyle f(x)=\sum_{n=0}^\infty a^n\cos(b^n\pi x)$.

Weierstrass showed that if

• $0,
• $b$ is an odd positive integer, and
• $\displaystyle ab>1+\frac32\pi$,

then $f$ is a continuous nowhere differentiable function. Hardy proved in 1916 that one can relax the conditions on $a,b$ to

• $0,
• $b>1$, and
• $ab\ge 1$.

Here, I just want to show some graphs, hopefully providing some intuition to help understand why we expect $f$ to be non-differentiable. The idea is that the cosine terms ensure that the partial sums  $\displaystyle f(m,x)=\sum_{n=0}^m a^n\cos(b^n\pi x)$, though smooth, have more and more “turns” on each interval as $m$ increases, so that in the limit, $f$ has “peaks” everywhere. Below is an animation (produced using Sage) comparing the graphs of $f(m,x)$ for $0\le m<20$ (and $-10\le x\le 10$), for $a=1/2$ and $b=11$, showing how the bends accumulate. (If the animations are not running, clicking on them solves the problem. As far as I can see, they do not work on mobiles.)

Below the fold, we show the same animation, zoomed in around $0$ by factors of $100$, $10^4$, and $10^6$, respectively, illustrating the fractal nature of $f$.