Suppose $X$ is a finite subset of the plane and for $0\leq \theta<\pi$, let $l\_\theta$ denote the line through the origin having angle $\theta$ with the positive $x$-axis. For how many values of $\theta$ can the projection $P\_{l\_\theta}$ of $X$ onto $l\_\theta$ have smaller order of magnitude? Say $|P\_{l\_\theta}(X)|\ll|X|^{1-\epsilon}$? For almost all $\theta$ this should not happen. Can it happen for more than a bounded number of $\theta$? 



Number of small projections
Yes; indeed the number of such $\theta$ can grow as $|X|^{1-2\epsilon}$,
which is unbounded for all $\epsilon < 1/2$. For large $N$
let $X$ be the set of integer points $(x,y)$ with $x^2 + y^2 < N$,
so $|X| \sim \pi N$.
If $l\_\theta$ has rational slope $a/b$ then $P\_{l\_\theta}(x,y)$
depends only on the integer $bx+ay$, which has absolute value at most
$\sqrt{a^2+b^2} \cdot N^{1/2}$ by Cauchy-Schwarz.
Therefore $|P\_{l\_\theta}(X)| \ll \sqrt{a^2+b^2} \cdot N^{1/2}$.
This is $\ll |X|^{1-\epsilon}$ if $a^2+b^2 \ll X^{1-2\epsilon}$,
and the number of such $\theta$ grows as $X^{1-2\epsilon}$, **QED**.


