In spite of a recent breakthrough on upper bounds of the size of cap sets (by Croot, Lev and Pach and by Ellenberg and Gijswijt), the classical cap set constructions had not been affected. In this work, we introduce a very different method of construction for caps in all affine spaces with odd prime modulus . Moreover, we show that for all primes with , the new construction leads to an exponentially larger growth of the affine and projective caps in and . For example, when , the existence of caps with growth follows from a three-dimensional example of Bose, and the only improvement had been to by Edel, based on a six-dimensional example. We improve this lower bound to .

We construct a unilateral lattice tiling of into hypercubes of two differnet side lengths or . This generalizes the Pythagorean tiling in . We also show that this tiling is unique up to symmetries, which proves a variation of a conjecture by Bölcskei from 2001. For positive integers and , this tiling also provides a tiling of .

It is proved that if is sufficiently large, then uniformly for all positive integers , we have where γ is the Euler constant. We also establish lower bounds for maximum of when and are fixed.