186 papers
This paper revisits the golden sieve to demonstrate its generation of Wythoff pairs from natural numbers, establishes its connection to hiccup sequences and Fraenkel's partitions when applied to arithmetic progressions, and introduces a new extraction sieve governed by affine transformations of hiccup parameters.
This paper establishes an isomorphism between the classical quasi-shuffle and the recently defined shuffle Hopf algebras for multiple zeta values by utilizing quasi-symmetric functions, while comparing this result to the well-known isomorphism by Hoffman, Newman, and Radford.
This paper introduces block-separated overpartitions, a constrained family where no two consecutive distinct part-blocks are both overlined, and demonstrates that their enumeration is governed by Fibonacci-type combinatorics, leading to a rich algebraic structure including Euler products and recurrence relations, while sharing the same exponential growth rate as ordinary partitions.
This paper characterizes product identities of the form among Hecke eigenforms over totally real number fields, proving that such identities occur exclusively for the field with exactly two instances, while also demonstrating their non-existence when both factors are Eisenstein series of distinct weights.
This paper advances Mazur's Program B for elliptic curves over by proving that the genus-69 modular curve has no non-CM rational points, a result achieved by linking these points to solutions of a generalized Fermat equation and reducing the complete classification of 7-adic Galois representations to finding rational points on a single plane quartic.
This paper defines a quaternionic analog of the classical Maass Spezialschar on split , characterizing this space of level one quaternionic modular forms via theta lifts from and period integrals, while also proposing and verifying a conjecture regarding the Dirichlet series of their standard -functions.
This paper establishes that for most parameter sets, the symmetry group of Markoff-type equations acts transitively on the majority of solutions modulo for a density one set of primes, with specific applications proving near-complete transitivity results for the -classification conjecture in and for solutions arising from generalized cluster algebras.
This paper introduces a notion of semistability for intrinsic reductions of non-archimedean rational functions at non-classical points in the Berkovich projective line and computes the resulting loci for quadratic rational iterations via a slope formula, demonstrating that these loci exhibit precise stationarity analogous to the polynomial case.
This paper combines additive combinatorics and Diophantine geometry to establish a uniform sum-product phenomenon for one-dimensional algebraic groups over , thereby resolving Bremner's conjecture on arithmetic progressions in elliptic curve coordinates and improving upon existing results by Bays--Breuillard regarding Elekes--Szabó-type theorems.
This paper explores the recursive structure of integer partitions by demonstrating how Sylvester waves, which represent partitions as sums of polynomial and quasiperiodic components, rely on integer weights that are themselves sums of partitions into smaller sets of integers.
This paper characterizes the monogenicity of irreducible trinomials of the form (where is prime) by classifying them according to their Galois groups, thereby extending the authors' previous research.
This paper classifies quadratic rational maps over with nonabelian automorphism groups that possess rational periodic points of periods 1, 2, and 3, proves the nonexistence of such points for periods 4 and 5, and establishes that the total number of rational preperiodic points is at most 6.
This paper constructs new infinite families of Ramanujan complexes with distinct local structures by utilizing super-definite unitary groups over totally real number fields, yielding novel examples across various types and providing a fully explicit rank-5 construction that generates golden gates for the real Lie group .
This paper establishes that for any ball in the unit square, the Hausdorff dimension of the intersection with the set of weighted badly approximable vectors equals that of the corresponding set of weighted approximable vectors, providing an explicit formula for this dimension in terms of the weight parameters.
This paper characterizes the spectrum of values for weak uniform Diophantine exponents associated with lattices in arbitrary dimensions.
Motivated by non-abelian Hodge theory, this paper formulates a Hodge-theoretic version of Grothendieck's anabelian conjecture by replacing Galois actions with natural -actions, proving an analog of Mochizuki's theorem for smooth projective hyperbolic curves over and extending the result to higher-dimensional ball quotient manifolds.
This paper proves that augmented admissible sets in Iwahori-Weyl groups are dual EL-shellable, thereby resolving a conjecture of Görtz and establishing the Cohen-Macaulayness of special fibers for local models with parahoric level structure—including previously open cases in residue characteristic 2 and non-reduced root systems—through a characteristic-free, intrinsic construction that yields an explicit shelling and inductive building procedure.
This paper resolves two longstanding problems from 1990 regarding indefinite integral ternary quadratic forms, highlighted by Margulis and initiated by Serre, by developing new tools to handle high ramification in sums weighted by diophantine invariants.
This paper investigates a new elementary method by Luo and Lin for solving quartic equations, successfully applying it to provide a straightforward solution for Bumby's equation $3X^4-2Y^2=1$ and proposing a conjecture that could extend this approach to an infinite family of similar equations.
This paper proves a conjecture by Michel and Venkatesh regarding the joinings of distinct Linnik problems through simultaneous quaternionic embeddings of imaginary quadratic fields, while also addressing a related non-equivariant conjecture by Aka, Einsiedler, and Shapira.