The puzzle was given a memorable nickname, the “happy ending” problem (or “happy end” problem as originally dubbed by Erdős), for reasons that had nothing to do with math. Instead, it reflected the primary nonmathematical consequence of their discussion of points, lines and shapes: Esther Klein and George Szekeres fell in love and married on June 13, 1937. Yet as the decades passed, mathematicians made virtually no progress in proving the conjecture. (The only other shape whose result is known is a hexagon, which requires at least 17 points, as proved by Szekeres and Lindsay Peters in 2006.) Now, in work recently published in the Journal of the American Mathematical Society, Andrew Suk of the University of Illinois, Chicago, provides nearly decisive evidence that the intuition that guided Erdős and Szekeres more than 80 years ago was correct.

A Puzzle of Clever Connections Nears a Happy End

# Liviu's Math Blog

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### On a class of random walks

I heard of these random walks from my number-theorist colleague Andrei Jorza who gave me a probabilistic translation of some observations in number theory. What follows is rather a variation on that theme.

There is a whole family of such walks, parametrized by a complex number $\newcommand{\ii}{\boldsymbol{i}}$ $\rho=\cos \theta +\ii \sin\theta$, $\theta\in [0,\pi]$, $\ii=\sqrt{-1}$.

This is discrete time walk on the configuration space $\newcommand{\lan}{\langle}$ $\newcommand{\ran}{\rangle}$

$$ \eC_\rho:= L[\rho] \times C(\rho,-\rho), $$

where $C(\rho,-\rho)$ is the subgroup of $S^1$ generated by $\pm\rho$, i.e.,

$$ C(\rho,-\rho)=\bigl\{\; \pm \rho^k;\;\;k\in\bZ \;\bigr\},$$

and $L[\rho]\subset \bC$ is the additive subgroup of $\bC$ generated by the elements $\rho^n$, $n\in\bZ_{\geq 0}$.

The state of the walk at time $n$ is given by a pair of random variables $(X_n, V_n)$, where one should think of $X_n\in L[\rho]$ as describing the "position'' and $V_n\in G(\rho,-\rho)$ as describing the ``velocity'' at time $n$.

Here is how the random walk takes place. Suppose that at time $n$ we are in the configuration $(X_n, V_n)$. To decide where to go next, toss a fair coin. If a the Head shows up then

$$V_{n+1}=\rho V_n,\;\; X_{n+1}= X_n+V_{n+1}. $$

Otherwise,

$$ V_{n+1}=-\rho V_n,\;\; X_{n+1}= X_n+V_{n+1}. $$

More abstractly, we are given a probability space $(\Omega,\eF,\bP)$ and a sequence of independent, identically distributed variables

$$ S_n:\Omega\to\{-1,1\},\;\;\bP[S_n=\pm 1]=\frac{1}{2},\;\;n\in\bZ_{\geq 0}. $$

If we set

$$

\whS_n:=\prod_{k=1}^nS_k, $$

then we have

$$

V_n= \whS_n \rho^n,\;\;X_n=\sum_{k=1}^n V_n=\sum_{k=1}^n \whS_k\rho_k.

$$

Observing that

$$\bE[\whS_k]=0,\;\;\var[\whS_k]=\prod_{j=1}^k\var[S_j]=1, $$ we deduce

$$\bE[X_n]=0,\;\;\var[S_n]=\bE[X_n\bar{X}_n] $$

$$=\sum_{j,k=1}^n \bE[ \whS_j\whS_k]\rho^{k-j}=\sum_{j=1}^n \bE[ \whS_j^2]+2\sum_{1\leq j<k\leq}^n \bE[ \whS_j\whS_k]\rho^{k-j}=n. $$

**Example 1.**Suppose that $\theta=\frac{\pi}{2}$ so that $\rho=\ii$. In this case $L[\rho]$ is the lattice $\bZ^2\subset \bC$ and $G(\ii,-\ii)$ is the group of forth order roots of $1$. At time $n=0$, the moving particle is at the origin facing East,. turns right or left (in this case North/South) with equal probability. Once reaches a new intersection, it turns right/left with equal probability. The following MAPLE generated animation a loop describes a 90-step portion of one such walk.

This is somewhat typical of what one observes in general.

**Example 2.**Suppose that $\theta =\frac{\pi}{3}$. In this case $G(\rho,-\rho)$ is the group of 6th order roots of $1$ and $L[\rho]$ is a lattice in $\bC$. The following MAPLE generated animation depicts a 90 step portion of such a walk and the patterns you observe are typical.

**Example 3.**Assume that the angle $\theta$ is small and $\frac{\theta}{\pi}$ is not rational, say $\cos \theta=0.95$. The animation below depicts a 90-step stretch of such a walk.

**Example 4.**For comparison, I have included an animation of a 90-step segment of the usual uniform random walk on $\bZ^2$.

Clearly, I am interested to figure out what is happening and I have a hard time asking a good question. A first question that comes to my mind would be to find statistical invariants that would distinguish the random walk in Example 1 from the random walk in Example 4. I will refer to them as $RW_1$ and $RW_4$

Clear each of these two walks surround many unit squares with vertices in the lattice $\bZ^2$. For $j=1,4$ will denote by $s_j(n)$ the expected number of unit squares surrounded by a walk of $RW_j$ of length $n$.. Which grows faster as $n\to\infty$, $s_1(n)$ or $s_4(n)$?

I will probably update this posting as I get more ideas, but maybe this shot in the dark will initiate a conversation and will suggest other, better questions.

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