Open-problem statements, with their sources attached.

Problem 10.5 (first part). Let $\mathbb{K}$ be a real number field. Then, for any $\varepsilon > 0$, there exists a lacunary sequence $(t_n)_{n \ge 1}$ of positive numbers in $\mathbb{K}$ such that $\limsup_{n \to \infty} \{\xi t_n\} \ge 1 - \varepsilon,$ for any real number $\xi$ not in $\mathbb{K}$.

Sendov's conjecture* states that for a polynomial $f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)$ with all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point.

There exists a Hadamard matrix for all $n = 4k$.

Let $A=\{1\leq a_1 < a_2 < \cdots\}$ and $B=\{1\leq b_1 < b_2 < \cdots\}$ be sets of integers with $a_n/b_n\to 1$.

If $A+B$ contains all sufficiently large positive integers then is it true that $\limsup 1_A\ast 1_B(n)=\infty$?

A conjecture of Erdős and Sárközy.

Inscribed rectangle problem* Does every Jordan curve admit inscribed rectangles of any given aspect ratio?

If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.

Problem 10.5 ("moreover" clause). With the same hypotheses as problem_10_5, the sequence $(t_n)$ can be chosen so that, for any real $\xi$ not in $\mathbb{K}$, each subinterval of $[0, 1]$ of length $\varepsilon$ contains a limit point of the sequence $(\{\xi t_n\})_{n \ge 1}$. This is strictly stronger than problem_10_5: the limsup bound is the special case at the subinterval $[1 - \varepsilon, 1]$.

The smallest order for which no Hadamard matrix is presently known is $668 = 4 * 167$.

Erdős Problem 1167.* Let $r \geq 2$ be finite, $\gamma \geq 2$, and $\lambda$ be an infinite cardinal. Let $\kappa_\alpha$ be cardinals for all $\alpha < \gamma$. Is it true that $2^\lambda \to (\kappa_\alpha + 1)_{\alpha < \gamma}^{r+1}$ implies $\lambda \to (\kappa_\alpha)_{\alpha < \gamma}^r?$ Here $+$ means cardinal addition, so that $\kappa_\alpha + 1 = \kappa_\alpha$ if $\kappa_\alpha$ is infinite.

A problem of Erdős, Hajnal, and Rado.

What is the smallest square that can contain 11 unit squares?

Reference: Wikipedia

Does every convex polygon have a vertex with no other 4 vertices equidistant from it?

Problem 10.6. Find a very rapidly increasing sequence $(m_n)_{n \ge 1}$ of positive integers such that $(\{\xi m_n\})_{n \ge 1}$ is dense modulo one for every irrational number $\xi$. Note: Furstenberg's $2^m3^n$ is sublacunary but requires two parameters.

Finite-target case.* When all $\kappa_\alpha$ are finite, $\kappa_\alpha + 1$ is the ordinary natural-number successor. Special case of erdos_1167.

What is the smallest square that can contain 17 unit squares?

Reference: Wikipedia

Erdős also conjectured that there is a $k$ for which every convex polygon has a vertex with no other $k$ vertices equidistant from it.

Problem 10.6, intermediate-growth variant.

Binary-color case.* The $\gamma = 2$ specialization (two color classes).

What is the smallest circle that can contain 3 unit squares?

Reference: Wikipedia

Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\to \infty$?

Problem 10.7. Let $\varepsilon$ be a positive real number. Are there arbitrarily large real numbers $\alpha$ such that $\alpha$ is not a Pisot number and all the fractional parts $\{\alpha^n\}$, $n \ge 1$, are lying in an interval of length $\varepsilon / \alpha$? [Bug12b]