Open-problem statements, with their sources attached.

Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?

Problem 4.1.* Let $\Omega \subset \mathbb{R}$ be a finite union of intervals and $\nu$ a weak tiling measure for $\Omega$. Must $\mathrm{supp}(\nu)$ have bounded density?

Problem 4.2.* Let $\Omega \subset \mathbb{R}$ be a finite union of three or more intervals. If $\Omega$ weakly tiles its complement, must it also tile its complement properly?

Problem 4.3.* Let $\Omega \subset \mathbb{R}$ be a finite union of intervals and $\nu$ a weak tiling measure for $\Omega$. Must $\nu$ be expressible as a convex combination of proper tiling measures?

Fuglede's conjecture* in one dimension: A bounded subset of ℝ with positive Lebesgue measure is spectral iff it tiles ℝ by translation.

Fuglede's conjecture* in two dimensions: A bounded subset of ℝ^2 with positive Lebesgue measure is spectral iff it tiles ℝ^2 by translation.

The Kakeya set conjecture: Kakeya sets in $\mathbb{R}^n$ have Hausdorff dimension $n$.