186 papers
This paper derives explicit formulae for the -dissection of a product of quintuple products when is a prime expressible as a sum of two squares, determines the sign patterns of the resulting Taylor series coefficients, and presents combinatorial applications of these findings.
This paper demonstrates that while BSD invariants themselves do not exhibit murmuration oscillations, the analytic order of the Tate-Shafarevich group uniquely modulates Frobenius trace murmurations and low-lying L-function zero distributions through a pure mean shift concentrated at small primes, revealing information not captured by other standard invariants.
This paper establishes that for any integer and any greater than the unique positive root of a specific polynomial, the restricted -fold sumset of a subset in a finite abelian group of sufficiently large odd order equals the entire group whenever , thereby generalizing previous results on cyclic groups and identifying as the optimal asymptotic density threshold.
This paper investigates the topological, metric, and fractal properties of the set of real numbers whose 4-adic digits possess existing frequencies and a specific asymptotic mean, providing an algorithm for constructing such points and establishing conditions for their Lebesgue measure and Hausdorff dimension.
This paper investigates the topological, metric, and fractal properties of subsets of the interval defined by specific asymptotic means of digits in their ternary representations, while exploring their relationship to numbers characterized by fixed digit frequencies.
This paper introduces the concept of the asymptotic mean of digits in the generalized -representation of real numbers and investigates the topological, metric, and fractal properties of sets defined by the existence or specific values of these means, while also exploring their connections to digit frequencies and fractal geometry.
This survey article reviews the theory of twisted dynamical zeta functions of Ruelle and Selberg and explores Fried's conjecture, drawing from a mini-course delivered at the Institut Henri Poincaré.
This paper unconditionally proves the quantum unique ergodicity conjecture for Pitale lifts on hyperbolic 4-manifolds by employing a novel amplification method with favorable geometric properties to overcome the challenges posed by non-temperedness.
This paper proves Kanade's previously open conjecture regarding dilogarithm identities and Nahm sums by leveraging Kirillov's identities and a dilogarithm ladder, while also proposing two new conjectures involving rank-2 matrices inspired by these results.
This paper extends previous results on the sum-product problem by proving that any set of 10 or 11 natural numbers must generate at least 30 or 34 distinct pairwise sums or products, respectively, while providing a classification of extremal sets and analyzing additive structures in small subsets of generalized geometric progressions.
This paper provides an elementary proof for Ramanujan-type identities expressing the squares of the Riemann zeta function at integer points through series involving hyperbolic functions, the digamma function, and Bernoulli numbers, while also correcting previous inaccuracies and adding new sections, a figure, and a reference.
This paper proves that the Halton sequence is not quasi-uniform in any dimension with pairwise relatively prime bases, and extends this result to disprove the quasi-uniformity of certain Halton-type sequences, including the Faure sequence.
This paper establishes that the number of rational points on an elliptic curve whose -coordinates lie in a -dimensional generalized arithmetic progression is bounded by a constant raised to the power of the curve's Mordell-Weil rank, a result derived by combining gap principles for large canonical heights with spherical code bounds and which implies restrictions on sets of points with small sumsets.
This paper demonstrates that the union of randomly digitally shifted Korobov polynomial lattice point sets achieves a star discrepancy with an inverse that depends linearly on the dimension, thereby narrowing the search for explicit constructions from a continuum to a finite set of candidates.
This paper introduces the concept of (W')-specification to prove that recurrence sets in a broad class of subshifts and piecewise expanding interval maps, which lack the standard specification property, possess full Hausdorff dimension.
This paper establishes elementary upper and lower bounds for the partition function and its generalizations by applying a geometric inequality in Euclidean space.
This paper constructs a modified universal Kolyvagin system for Hida families of modular forms using big Heegner points on Shimura curves, generalizing previous work to a quaternionic setting without the classical Heegner hypothesis and proving one divisibility of the anticyclotomic Iwasawa main conjecture.
This paper computes the first and second moments of sums of normalised Hecke eigenvalues over holomorphic cusp forms of large weight, demonstrating that in the range , the second moment scales between and , a sharp contrast to the linear growth observed in the lower range .
This paper proves a Gersten-type conjecture for étale sheaves, including étale logarithmic Hodge-Witt sheaves and -adic Tate twists, over henselian local rings of normal crossing varieties in positive characteristic, and applies this result to establish a relative version of the conjecture for -adic étale Tate twists over semistable families in mixed characteristic as well as a generalization of Artin's theorem on Brauer groups.
This paper introduces the concept of abelian-normal numbers by adapting abelian complexity functions to digit expansions, constructs a non-normal analogue of Champernowne's constant that satisfies this new property, and proposes open problems regarding its behavior.