= Compact Kakeya construction from small projections
{title2=$B=\{(t,y+tx):(x,y)\in K,\ t\in[0,1]\}$}
Choose a compact $K\subset[0,1]\times\mathbb R$ with full horizontal projection and zero-length projections $y+tx$ for $t\in[0,1]$. Then the displayed set is compact, has zero area by the <Fubini theorem>, and contains a segment of every slope in $[0,1]$. Such parameter sets exist by the <Baire category theorem> in the <Hausdorff metric>: fine parallel segments approximate any parameter set while making a fixed band of projections uniformly small. Finite grids and a countable intersection make every required projection null. Four rotated copies supply every planar direction.
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