How many points can be placed in an n & times;n$n imes n$ grid so that every (affine) line contains at most k$k$ points? We prove that for n >= k >= 1037$n \geqslant k \geqslant 10<^>{37}$ the maximum number of points is exactly kn$kn$ . Our proof builds on the recent work of Kov & aacute;cs, Nagy and Szab & oacute; (who proved an analogous result when k$k$ is at least about nlogn$\sqrt {n \log n}$ ), incorporating ideas of Jain and Pham. Using the same approach, we also obtain new bounds for higher-dimensional extensions of this problem.
We show that the elements of the Kazhdan-Lusztig basis of the spherical Hecke algebra of type have an atomic decomposition. As a by-product, we obtain a new algorithm to compute generalized Kostka-Foulkes polynomials in type .
The DGA of planar loops is a pictorial chain complex introduced in a recent paper of the author, Boyd, Randal-Williams and Sroka, where a minimal model for it was given. In the first non-trivial case, , we give a new model which incorporates two natural 'reflection' involutions, as well as a more explicit description of the existing model.
In this paper, we provide an alternative proof of Chandee and Li's result on the second moment of special -values. Our method is conceptually more direct as it neither detects the "Eisenstein-Kloosterman" cancelation nor uses the Poisson summation formula.
An approximate Hadamard matrix is a well-conditioned square matrix with all entries in {+/- 1}. We measure the quality of a matrix by its condition number, that is, the ratio of its largest and smallest singular values. We prove that for any fixed positive alpha < 17/92, every sufficiently large dimension admits an approximate Hadamard matrix with condition number at most 1 + n(-alpha). In particular, the smallest possible condition number tends to 1 as n -> infinity. Conversely, there exists an absolute constant c > 0 such that for every sufficiently large n (sic) 0 (mod 4), every n & times; n matrix with entries in {+/- 1} has condition number at least 1 + c(log n)/n. Along the way, we resolve a problem of Jaming and Matolcsi concerning flat orthogonal matrices, and we conclude by describing several explicit infinite families of approximate Hadamard matrices.
