More Like this

1. Lower bounds on Bourgain’s constant for harmonic measure

   https://doi.org/10.4153/S0008414X2300069X  

   Badger, Matthew; Genschaw, Alyssa (December 2024, Canadian Journal of Mathematics)

   Abstract For every$$n\geq 2$$, Bourgain’s constant$$b_n$$is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most$$n-b_n$$for every domain in$$\mathbb {R}^n$$on which harmonic measure is defined. Jones and Wolff (1988,Acta Mathematica161, 131–144) proved that$$b_2=1$$. When$$n\geq 3$$, Bourgain (1987,Inventiones Mathematicae87, 477–483) proved that$$b_n>0$$and Wolff (1995,Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton University Press, Princeton, NJ, 321–384) produced examples showing$$b_n<1$$. Refining Bourgain’s original outline, we prove that$$\begin{align*}b_n\geq c\,n^{-2n(n-1)}/\ln(n),\end{align*}$$for all$$n\geq 3$$, where$$c>0$$is a constant that is independent ofn. We further estimate$$b_3\geq 1\times 10^{-15}$$and$$b_4\geq 2\times 10^{-26}$$. 

   more » « less
2. A spatial mutation model with increasing mutation rates

   https://doi.org/10.1017/jpr.2022.120  

   Chao, Brian; Schweinsberg, Jason (December 2023, Journal of Applied Probability)

   Abstract We consider a spatial model of cancer in which cells are points on thed-dimensional torus$$\mathcal{T}=[0,L]^d$$, and each cell with$$k-1$$mutations acquires akth mutation at rate$$\mu_k$$. We assume that the mutation rates$$\mu_k$$are increasing, and we find the asymptotic waiting time for the first cell to acquirekmutations as the torus volume tends to infinity. This paper generalizes results on waiting for$$k\geq 3$$mutations in Fooet al.(2020), which considered the case in which all of the mutation rates$$\mu_k$$are the same. In addition, we find the limiting distribution of the spatial distances between mutations for certain values of the mutation rates. 

   more » « less
3. p -ADIC L -FUNCTIONS AND RATIONAL POINTS ON CM ELLIPTIC CURVES AT INERT PRIMES

   https://doi.org/10.1017/S147474802300021X  

   Burungale, Ashay A; Kobayashi, Shinichi; Ota, Kazuto (May 2024, Journal of the Institute of Mathematics of Jussieu)

   Abstract LetKbe an imaginary quadratic field and$$p\geq 5$$a rational prime inert inK. For a$$\mathbb {Q}$$-curveEwith complex multiplication by$$\mathcal {O}_K$$and good reduction atp, K. Rubin introduced ap-adicL-function$$\mathscr {L}_{E}$$which interpolates special values ofL-functions ofEtwisted by anticyclotomic characters ofK. In this paper, we prove a formula which links certain values of$$\mathscr {L}_{E}$$outside its defining range of interpolation with rational points onE. Arithmetic consequences includep-converse to the Gross–Zagier and Kolyvagin theorem forE. A key tool of the proof is the recent resolution of Rubin’s conjecture on the structure of local units in the anticyclotomic$${\mathbb {Z}}_p$$-extension$$\Psi _\infty $$of the unramified quadratic extension of$${\mathbb {Q}}_p$$. Along the way, we present a theory of local points over$$\Psi _\infty $$of the Lubin–Tate formal group of height$$2$$for the uniformizing parameter$$-p$$. 

   more » « less
4. Efficient simplicial replacement of semialgebraic sets

   https://doi.org/10.1017/fms.2023.36  

   Basu, Saugata; Karisani, Negin (January 2023, Forum of Mathematics, Sigma)

   Abstract Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set$$S \subset \mathbb {R}^k$$by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex$$\Delta $$, whose geometric realization,$$|\Delta |$$, is semialgebraically homeomorphic toS. In this paper, we consider a weaker version of this question. We prove that for any$$\ell \geq 0$$, there exists an algorithm which takes as input a description of a semialgebraic subset$$S \subset \mathbb {R}^k$$given by a quantifier-free first-order formula$$\phi $$in the language of the reals and produces as output a simplicial complex$$\Delta $$, whose geometric realization,$$|\Delta |$$is$$\ell $$-equivalent toS. The complexity of our algorithm is bounded by$$(sd)^{k^{O(\ell )}}$$, wheresis the number of polynomials appearing in the formula$$\phi $$, andda bound on their degrees. For fixed$$\ell $$, this bound issingly exponentialink. In particular, since$$\ell $$-equivalence implies that thehomotopy groupsup to dimension$$\ell $$of$$|\Delta |$$are isomorphic to those ofS, we obtain a reduction (having singly exponential complexity) of the problem of computing the first$$\ell $$homotopy groups ofSto the combinatorial problem of computing the first$$\ell $$homotopy groups of a finite simplicial complex of size bounded by$$(sd)^{k^{O(\ell )}}$$. 

   more » « less
5. Bipartite-ness under smooth conditions

   https://doi.org/10.1017/S0963548323000019  

   Jiang, Tao; Longbrake, Sean; Ma, Jie (February 2023, Combinatorics, Probability and Computing)

   Abstract Given a family$$\mathcal{F}$$of bipartite graphs, theZarankiewicz number$$z(m,n,\mathcal{F})$$is the maximum number of edges in an$$m$$by$$n$$bipartite graph$$G$$that does not contain any member of$$\mathcal{F}$$as a subgraph (such$$G$$is called$$\mathcal{F}$$-free). For$$1\leq \beta \lt \alpha \lt 2$$, a family$$\mathcal{F}$$of bipartite graphs is$$(\alpha,\beta )$$-smoothif for some$$\rho \gt 0$$and every$$m\leq n$$,$$z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$$. Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any$$(\alpha,\beta )$$-smooth family$$\mathcal{F}$$, there exists$$k_0$$such that for all odd$$k\geq k_0$$and sufficiently large$$n$$, any$$n$$-vertex$$\mathcal{F}\cup \{C_k\}$$-free graph with minimum degree at least$$\rho (\frac{2n}{5}+o(n))^{\alpha -1}$$is bipartite. In this paper, we strengthen their result by showing that for every real$$\delta \gt 0$$, there exists$$k_0$$such that for all odd$$k\geq k_0$$and sufficiently large$$n$$, any$$n$$-vertex$$\mathcal{F}\cup \{C_k\}$$-free graph with minimum degree at least$$\delta n^{\alpha -1}$$is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families$$\mathcal{F}$$consisting of the single graph$$K_{s,t}$$when$$t\gg s$$. We also prove an analogous result for$$C_{2\ell }$$-free graphs for every$$\ell \geq 2$$, which complements a result of Keevash, Sudakov and Verstraëte. 

   more » « less