Five miniatures

Oddtown

Odd distances

Triangle regions

Rounding trouble

Equiangular lines

Oddtown

Oddtown has $n$ residents

1. Every club must have an odd number of members
2. Any two clubs must have an even number of common members

What is the maximum number of clubs?

$A$$A^T$$ = \begin{bmatrix} - & \mathbf{a_1} & - \\ - & \mathbf{a_2} & - \\ & \vdots & \\ - & \mathbf{a_m} & - \end{bmatrix}$$\begin{bmatrix} | & | & & | \\ \mathbf{a_1} & \mathbf{a_2} & \dots & \mathbf{a_m} \\ | & | & & | \end{bmatrix}$

$$= \begin{bmatrix}|C_1| & |C_1 \cap C_2| & \dots & |C_1 \cap C_m| \\ |C_2 \cap C_1| & |C_2| & \dots & |C_2 \cap C_m| \\ \vdots & \vdots & \ddots & \vdots \\ |C_m \cap C_1| & |C_m \cap C_2| & \dots & |C_m| \end{bmatrix}$$

$$M := AA^T \equiv \begin{bmatrix}1 & & & \\ & 1 & & \\ & & \ddots & \\ & & & 1\end{bmatrix}$$

$\det(M) \equiv 1$

$m =$ $\mathrm{rank}(M)$ $\le$ $\mathrm{rank}(A)$ $ \le n$

Even Town has $n$ residents

1. Every club must have an even number of members
2. Any two clubs must have an even number of common members

What is the maximum number of distinct clubs?

Five miniatures

Oddtown

Odd distances

Triangle regions

Rounding trouble

Equiangular lines

Odd distances

Can the plane contain three points whose pairwise distances are all odd?

Can the plane contain four points whose pairwise distances are all odd?

Suppose such four points exist: $(0,0) = $ $\mathbf{p_0}, \mathbf{p_1}, \mathbf{p_2}, \mathbf{p_3} \in \mathbb{R}^2$

$A$$A^T$$ = \begin{bmatrix} - & \mathbf{p_1} & - \\ - & \mathbf{p_2} & - \\ - & \mathbf{p_3} & - \end{bmatrix}$$\begin{bmatrix} | & | & | \\ \mathbf{p_1} & \mathbf{p_2} & \mathbf{p_3} \\ | & | & | \end{bmatrix}$

$$= \begin{bmatrix}\mathbf{p_1} \cdot \mathbf{p_1} & \mathbf{p_1}\cdot\mathbf{p_2} & \mathbf{p_1}\cdot\mathbf{p_3} \\ \mathbf{p_2}\cdot\mathbf{p_1} & \mathbf{p_2}\cdot\mathbf{p_2} & \mathbf{p_2}\cdot\mathbf{p_3} \\ \mathbf{p_3}\cdot\mathbf{p_1} & \mathbf{p_3}\cdot\mathbf{p_2} & \mathbf{p_3}\cdot\mathbf{p_3} \end{bmatrix}$$

$$|\mathbf{p_1} - \mathbf{p_2}|^2 = |\mathbf{p_1}|^2 + |\mathbf{p_2}|^2 - 2\mathbf{p_1}\cdot\mathbf{p_2}$$

$$2\mathbf{p_1}\cdot\mathbf{p_2} \equiv 1$$

$M := 2AA^T \equiv \begin{bmatrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$\implies \det(M) \equiv 0$

What now?

$$2AA^T = \begin{bmatrix}2\mathbf{p_1} \cdot \mathbf{p_1} & 2\mathbf{p_1}\cdot\mathbf{p_2} & 2\mathbf{p_1}\cdot\mathbf{p_3} \\ 2\mathbf{p_2}\cdot\mathbf{p_1} & 2\mathbf{p_2}\cdot\mathbf{p_2} & 2\mathbf{p_2}\cdot\mathbf{p_3} \\ 2\mathbf{p_3}\cdot\mathbf{p_1} & 2\mathbf{p_3}\cdot\mathbf{p_2} & 2\mathbf{p_3}\cdot\mathbf{p_3} \end{bmatrix}$$

$\mathbf{p_1} \cdot \mathbf{p_1} =$ square of an odd number $\equiv 1 \pmod{8}$

$2\mathbf{p_1} \cdot \mathbf{p_2} = |\mathbf{p_1}|^2 + |\mathbf{p_2}|^2 - |\mathbf{p_1} - \mathbf{p_2}|^2$ $\equiv 1 \pmod{8}$

$$2AA^T \equiv \begin{bmatrix}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}$$

$$\implies \det(2AA^T) \equiv 4 \pmod{8}$$

$3 =$ $\mathrm{rank}(M)$ $\le$ $\mathrm{rank}(A)$ $ \le 2$

Problem

What is the maximum number of points in space whose pairwise distances are all odd?

Five miniatures

Oddtown

Odd distances

Triangle regions

Rounding trouble

Equiangular lines

Triangle regions

$n$ lines in the plane (no three concurrent, no two parallel)

How many triangular regions must there be? At least $n-2$?

Observation: if we move the lines slightly, the regions barely change

Fix two lines. Can we translate the other $n-2$ lines
while preserving all triangular regions?

Preserving triangular regions $\Leftrightarrow$ the line velocities satisfy linear constraints

Proof by contradiction: suppose there are $< n-2$ triangular regions

Number of constraints $<$ number of unknowns (velocities)

$\implies$ there is a nonzero solution, so the lines can move at constant velocity!

The first moment when three lines become concurrent?

Before the first triple intersection, every region is essentially unchanged

But just before that moment...

Five miniatures

Oddtown

Odd distances

Triangle regions

Rounding trouble

Equiangular lines

Rounding trouble

An online shop is processing orders.

Suddenly all coins worth less than one dollar are abolished!

How should the shop round each item price

so that every order total changes only a little?

Suppose the items are $1, 2, \dots, n$, and the fractional part of item $i$ is $c_i$

Orders $S_1, \dots, S_m \subset \{1, 2, \dots, n\}$

Beck-Fiala theorem

If each $i$ appears in at most $d$ sets $S_k$,

then there are $z_1, \dots, z_n \in \{0,1\}$

such that $\left|\sum_{i \in S_k} z_i - \sum_{i \in S_k}c_i\right| < d$ for every $S_k$.

Let the prices move?

$c_i \in (0,1) \to x_i \in [0,1] \to z_i \in \{0,1\}$

Rule: once $x_i$ reaches 0 or 1, it stays fixed

Before it is fixed, we say $x_i$ is floating

While prices are floating, which $S_k$ should worry us?

Call an order dangerous if it contains more than $d$ floating items

Fewer and fewer items float, so fewer and fewer orders are dangerous

How can prices move while every dangerous order total stays fixed?

Dangerous orders $\Leftrightarrow$ linear constraints on the floating prices $x_i$

Number of constraints $<$ number of floating prices?

Nonzero solution $\implies$ floating prices can move at constant velocity

until some floating price first reaches 0 or 1

After an order stops being dangerous,
its at most $d$ floating items can change the total by $< d$.

Discrepancy theory

$S_1, \dots, S_m \subset \{1,2,\dots, n\}$

Given a coloring $f\colon \{1,2,\dots, n\} \to \{-1, 1\}$

Score it by $\chi(f) = \max_k |\sum_{i\in S_k}f(i)|$

If each $i$ appears in at most $d$ sets $S_k$,

how large can $\min_f\chi(f)$ be?

Five miniatures

Oddtown

Odd distances

Triangle regions

Rounding trouble

Equiangular lines

Equiangular lines

How many lines can we place in the plane so that all pairwise angles are equal?

How many equiangular lines can we place in space?

3 lines? 4 lines? 6 lines?

Choose unit column vectors $v_1, \dots, v_n \in \mathbb{R}^3$ on the equiangular lines

$v_iv_i^T$ is a 3 x 3 symmetric matrix

What is the dimension of the space of 3 x 3 symmetric matrices? 6!

Are the symmetric matrices $v_1v_1^T, \dots, v_nv_n^T$ linearly independent?

Suppose $a_1v_1v_1^T + \dots + a_nv_nv_n^T = 0$

$\implies v_1^T(a_1v_1v_1^T + \dots + a_nv_nv_n^T)v_1 = 0$

$\implies a_1 + a_2 \alpha^2 + \dots + a_n \alpha^2 = 0$

$\implies (1-\alpha^2)a_1 + \alpha^2(a_1 + \dots + a_n) = 0$

$\implies \dots \implies a_1 = \dots = a_n = 0$

The equiangular lines problem

In $d$ dimensions, let $N_\alpha(d)$ be the maximum number of equiangular lines with angle $\arccos\alpha$