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Assume for simplicity that sets $A_i\subset\mathbb{R}$ are compact. If $A_1$ and $A_2$ have positive length, then $A_1+A_2$ contains an interval. That is a variant of the classical Steinhaus theorem ...

Suppose $\sum_{\xi \in \mathbb{Z}^d}{a_{\xi}}e^{i\langle x, \xi\rangle }$ converges spherically pointwise to $0$ for all $x \in \mathbb{T}^d$, i.e. $\lim_{R \to \infty} \sum_{|\xi| < R}{a_{\xi}}e^{...

in the paper, ON KAKEYA MAPS WITH REGULARITY ASSUMPTIONS, the author of the paper consider the Kakeya conjecture with some regularity on Kakeya map, Kekeya conjecture[Hausdorff dimension version] If ...