  Quantitative formulations of Fefferman's counterexample for the ball
multiplier are naturally linked to square function estimates for conical and
directional multipliers. In this article we develop a novel framework for these
square function estimates, based on a directional embedding theorem for
Carleson sequences and multi-parameter time-frequency analysis techniques. As
applications we prove sharp or quantified bounds for Rubio de Francia type
square functions of conical multipliers and of multipliers adapted to
rectangles pointing along $N$ directions. A suitable combination of these
estimates yields a new and currently best-known logarithmic bound for the
Fourier restriction to an $N$-gon, improving on previous results of A. Cordoba.
Our directional Carleson embedding extends to the weighted setting, yielding
previously unknown weighted estimates for directional maximal functions and
singular integrals.
