We establish a matching lower bound for the sextic large sieve over the Eisenstein integers in an unbalanced range, with both indices restricted to squarefree ideals. Let $\Xi_6(A,B)$ denote the squared operator norm of the sextic residue-character matrix with row and column norms at most $A$ and $B$, respectively, outside a fixed finite set of primes containing those above $6$. For every fixed $0<\delta<3/11$ and every $\varepsilon>0$, we prove $\Xi_6(A,B)\gg_{\delta,\varepsilon}(AB)^{-\varepsilon}A^{5/6}B^{1/3}$ for $B^{16/11+\delta}\le A\le B^{2-\delta}$. Consequently, the full-cutoff norm satisfies $\Xi_6(X,X)\gg_\varepsilon X^{17/16-\varepsilon}$. The main difficulty is to preserve the large second moment supplied by Gauss sums when the column moduli are restricted to be squarefree. We prove a second-moment formula uniform in the prime exclusions and character twists arising from M\"obius inversion, and use it to show that the squarefree projection retains a positive proportion of the original moment. A separate argument combining cubic and sextic large-sieve bounds then transfers the lower bound to squarefree rows.
Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities $1,2,4,\ldots,2^{m-1}$, normalized by $2^m-1$. We prove the conjecture through a dimension-free intermediate statement: every such partition contains a component of density a...
We study the microscopic spectrum at the origin of ratios of independent complex Girko matrices. Under bounded-density and finite-moment assumptions, together with circular second-moment matching, we prove compact-uniform convergence of the normalized determinants to a random entire function constructed from the comple...
We determine the connected geometric monodromy of the top-weight sheaves attached to linearly constrained Kloosterman families in every dimension n>= 4 and every characteristic p>max(2,n). The quantitative input is a fourth-moment identity whose auxiliary geometry depends on the moment degree rather than on the ambient...
In the setting of the previously reported degree-seven counterexample of ani, we present an explicit monic polynomial with real rational coefficients and a direct proof by certified separators. The polynomial has seven simple zeros in the open unit disk. Every continuous path joining distinct zeros in its closed unit s...
We give a proof of the existence of two-sided product expanding codes which, unlike the earlier result of Kalachev and Panteleev (FOCS, 2025), does not rely on explicit constructions of asymptotically optimal locally testable codes ($c^3$-LTCs). For every fixed number of component codes and dimensions whose rates are b...
Eshan Chattopadhyay, Noam Ringach, Nicholas Spooner· 0 citations
Let X be a centered infinitely divisible random vector with finite second moment and covariance matrix Sigma. We define the radial pole C_X(t) as the smallest scale in a right sub-gamma bound forwhose quadratic proxy is fixed at the true variance t^T Sigma t. A canonical directional measure and an independent Beta(1,2)...
Yi-Chuan Chen, Xin Wang· 1 citation· ⚡1
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