Let λ \lambda be the Liouville function, defined as λ ( n ) ≔ ( − 1 ) Ω ( n ) \lambda (n) ≔(-1)^{\Omega (n)} where Ω ( n ) \Omega (n) is the number of prime factors of n n with multiplicity. In 2021, Helfgott and Radziwiłł proved that ∑ n ⩽ x 1 n λ ( n ) λ ( n + 1 ) ≪ log x ( log log x ) 1 / 2 , \begin{equation*} \sum _{n\leqslant x} \frac {1}{n}\lambda (n) \lambda (n+1) \ll \frac {\log x}{(\log \log x)^{1/2}}, \end{equation*} improving earlier results by Tao and Teräväinen. We prove that ∑ n ⩽ x 1 n λ ( n ) λ ( n + 1 ) ≪ ( log x ) 1 − c \begin{equation*} \sum _{n\leqslant x} \frac 1n \lambda (n) \lambda (n+1) \ll (\log x)^{1-c} \end{equation*} for some absolute constant c > 0 c>0 . This appears to be best possible with current methods.
An error is noted in the averaging argument in “Convergence of exclusion processes and the KPZ equation to the KPZ fixed point”, J. Amer. Math. Soc. 36 (2023), 251-289. Consequently, the method proves tightness for non-nearest neighbor exclusion processes under the 1 : 2 : 3 1:2:3 scaling, but convergence to the KPZ fixed point only for perturbations of TASEP vanishing in the limit.
Transitivity properties of the group of morphisms generated by Vieta involutions on the solutions to the Markoff equation modulo primes are established, yielding forms of strong approximation for the Markoff surface. These are applied to show that almost all Markoff numbers are highly divisible.
A long-standing conjecture of De Giorgi asserts that every monotone solution of the Allen–Cahn equation in R n + 1 \mathbb {R}^{n+1} is one-dimensional if n ≤ 7 n \leq 7 . A stronger version of the conjecture, also widely studied and often called “the stable De Giorgi conjecture”, proposes that every stable solution in R n \mathbb {R}^n must be one-dimensional for n ≤ 7 n \leq 7 . To this date, both conjectures remain open for 3 ≤ n ≤ 7 3 \leq n \leq 7 . An elegant variant of this problem, advocated by Caffarelli, Córdoba, and Jerison since the 1990s, considers a free boundary version of the Allen–Cahn equation. This variant features a step-like double-well potential, leading to multiple free boundaries. Locally, near each free boundary, the solution satisfies the Bernoulli free boundary problem. However, the interaction of the free boundaries causes the global behavior of the solution to resemble that of the Allen–Cahn equation. In this paper, we establish the validity of the stable De Giorgi conjecture in dimension 3 3 for the free boundary Allen–Cahn equation and, as a corollary, we prove the corresponding De Giorgi conjecture for monotone solutions in dimension 4 4 . To obtain these results, a key aspect of our work is to address a classical open problem in free boundary theory of independent interest: the classification of global stable solutions to the one-phase Bernoulli problem in three dimensions. This result, which is the core of our paper, implies universal curvature estimates for local stable solutions to Bernoulli, and serves as a foundation for adapting some classical ideas from minimal surface theory—after significant refinements—to the free boundary Allen–Cahn equation.
In this paper and its sequel, we develop a technique for controlling the distribution of l∞-Selmer groups in degree l twist families of Galois modules over number fields. Given an elliptic curve E over a number field satisfying certain technical conditions, this technique can be used to show that 100% of the quadratic twists of E have rank at most 1. It also can be used to prove distributional results about the l∞-class groups in the family of degree l cyclic extensions of a given number field. For this work, we develop the theory of the fixed point Selmer group, which serves as the base layer of the l∞-Selmer group. This paper gives a technique for finding the distribution of l∞-Selmer groups in certain families of twists where the fixed point Selmer group is stable. In the sequel paper, we will give a technique for controlling fixed point Selmer groups.
