Worked solution: The planar case: every Kakeya set in $\mathbb{R}^2$ has dimension 2 (Davies, 1971)

Besicovitch (1919) constructed a compact set $E\subset\mathbb{R}^2$ of Lebesgue measure zero containing a unit line segment in every direction. The construction — 'Perron trees', sliding triangles so that copies overlap massively — shows the zero-measure conclusion survives even for sets in which a unit segment moves continuously through all directions (Kakeya's original needle problem).

Measure zero means: for every $\epsilon>0$, $E$ can be covered by discs of total area $<\epsilon$. Yet $E$ contains uncountably many full-length segments. So two-dimensional Lebesgue measure — the naive yardstick of 'how much room a set occupies' — completely fails to register the structure forcing every direction to be present. This is exactly the paradox that makes the question interesting: the set is simultaneously negligible (in area) and maximal (in directional content).

The escape is to change the invariant rather than the question. Hausdorff and Minkowski dimensions measure how covering numbers scale with resolution, not raw area, so a measure-zero set can still be forced to have full dimension $2$. Davies's theorem is precisely that statement; the rest of the proof builds the machinery (tube discretisation, overlap control, Cauchy–Schwarz) that makes it quantitative.