Abstract
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 modulusp. 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 .
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