New logarithm laws and lattice point bounds yield a proof of power loss in the Mizohata-Takeuchi conjecture with explicit errors and establish genericity in C^k.
Successive Minima and Lattice Points
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abstract
The main purpose of this note is to prove an upper bound on the number of lattice points of a centrally symmetric convex body in terms of the successive minima of the body. This bound improves on former bounds and narrows the gap towards a lattice point analogue of Minkowski's second theorem on successive minima. Minkowski's proof of his second theorem is rather lengthy and it was also criticised as obscure. We present a short proof of Minkowski's second theorem on successive minima, which, however, is based on the ideas of Minkowski's proof.
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Cusp Excursions, Lattice Points on Manifolds, and the Mizohata-Takeuchi Conjecture
New logarithm laws and lattice point bounds yield a proof of power loss in the Mizohata-Takeuchi conjecture with explicit errors and establish genericity in C^k.