标题： 加权$\ell^2$中的阈值通过稀疏傅里叶谱 显示英文标题

标题： Thresholds in weighted $\ell^2$ via sparse Fourier spectrum

摘要： 我们研究加权$\ell^2$空间中的阈值现象，并表征了可以构造具有小支撑或最大坏范围的元素的关键区域。 我们的结果在多个方面被证明是最佳的，且我们的证明主要依赖于涉及稀疏傅里叶谱的技术。 我们进一步表明，这些看似病态的构造实际上从某些范畴论的角度来看是普遍的。 显示英文摘要

摘要： We investigate threshold phenomena in weighted $\ell^2$-spaces and characterize the critical regimes where elements with either small support or maximally bad range can be constructed. Our results are shown to be optimal in several respects, and our proofs principally rely on techniques involving sparse Fourier spectrum. We further show that these seemingly pathological constructions are actually generic from certain categorical perspectives.