6 problems

LLL mixing and density conjecture. (i) The sequence $(|\mu_{k(i)}|)_{i=1,2,\ldots}$ is strongly mixing as a stochastic process. (ii) Each $|\mu_{k(i)}|$ is contained in a compact s…

Let ${\mathbf v}=[v_1,\dots,v_n]$ be a modular symbol with $\|{\mathbf v}\|>1$, and define … Let $E$ be the standard basis of ${\mathbb Z}^n$. The quadratic form $b({\mathbf v})$ i…

Let $k\geq 4$, and let $\delta^*(k)$ denote the optimal balanced discrepancy for the module-lattice sign-selection problem. Write $\delta^*$ for the constant value proposed by the…

Bounded output-density conjecture. For every such $\mathcal{D}$, the probability density function of $\mathcal{D}^{\circ}$ is bounded from above by a constant $C$ that depends only…

Linear speed-up conjecture. When the number of processors is at most

Let $\mathcal{D}$ be a distribution on the set of bases of $\mathbb{R}^n$, and let $\mathcal{D}^\circ$ denote the corresponding output distribution of LLL or LLL-SP. Bounded output…