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1. Increasing subsequences, matrix loci and Viennot shadows

   https://doi.org/10.1017/fms.2024.75  

   Rhoades, Brendon (January 2024, Forum of Mathematics, Sigma)

   Abstract Let$${\mathbf {x}}_{n \times n}$$be an$$n \times n$$matrix of variables, and let$${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$$be the polynomial ring in these variables over a field$${\mathbb {F}}$$. We study the ideal$$I_n \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$$generated by all row and column variable sums and all products of two variables drawn from the same row or column. We show that the quotient$${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$$admits a standard monomial basis determined by Viennot’s shadow line avatar of the Schensted correspondence. As a corollary, the Hilbert series of$${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$$is the generating function of permutations in$${\mathfrak {S}}_n$$by the length of their longest increasing subsequence. Along the way, we describe a ‘shadow junta’ basis of the vector space ofk-local permutation statistics. We also calculate the structure of$${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$$as a graded$${\mathfrak {S}}_n \times {\mathfrak {S}}_n$$-module. 

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2. Intersective sets for sparse sets of integers

   https://doi.org/10.1017/etds.2024.73  

   BIENVENU, PIERRE-YVES; GRIESMER, JOHN T; LE, ANH N; LÊ, THÁI HOÀNG (May 2025, Ergodic Theory and Dynamical Systems)

   Abstract For$$E \subset \mathbb {N}$$, a subset$$R \subset \mathbb {N}$$isE-intersectiveif for every$$A \subset E$$having positive relative density,$$R \cap (A - A) \neq \varnothing $$. We say thatRischromatically E-intersectiveif for every finite partition$$E=\bigcup _{i=1}^k E_i$$, there existsisuch that$$R\cap (E_i-E_i)\neq \varnothing $$. When$$E=\mathbb {N}$$, we recover the usual notions of intersectivity and chromatic intersectivity. We investigate to what extent the known intersectivity results hold in the relative setting when$$E = \mathbb {P}$$, the set of primes, or other sparse subsets of$$\mathbb {N}$$. Among other things, we prove the following: (1) the set of shifted Chen primes$$\mathbb {P}_{\mathrm {Chen}} + 1$$is both intersective and$$\mathbb {P}$$-intersective; (2) there exists an intersective set that is not$$\mathbb {P}$$-intersective; (3) every$$\mathbb {P}$$-intersective set is intersective; (4) there exists a chromatically$$\mathbb {P}$$-intersective set which is not intersective (and therefore not$$\mathbb {P}$$-intersective). 

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3. Galois groups and prime divisors in random quadratic sequences

   https://doi.org/10.1017/S0305004123000439  

   DOYLE, JOHN R.; HEALEY, VIVIAN OLSIEWSKI; HINDES, WADE; JONES, RAFE (January 2024, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract Given a set$$S=\{x^2+c_1,\dots,x^2+c_s\}$$defined over a field and an infinite sequence$$\gamma$$of elements ofS, one can associate an arboreal representation to$$\gamma$$, generalising the case of iterating a single polynomial. We study the probability that a random sequence$$\gamma$$produces a “large-image” representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most setsSdefined over$$\mathbb{Z}[t]$$, and we conjecture a similar positive-probability result for suitable sets over$$\mathbb{Q}$$. As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify allSpossessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration. 

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4. 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$$-p$$. 

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5. Almost all orbits of the Collatz map attain almost bounded values

   https://doi.org/10.1017/fmp.2022.8  

   Tao, Terence (January 2022, Forum of Mathematics, Pi)

   Abstract Define theCollatz map$${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$$on the positive integers$$\mathbb {N}+1 = \{1,2,3,\dots \}$$by setting$${\operatorname {Col}}(N)$$equal to$$3N+1$$whenNis odd and$$N/2$$whenNis even, and let$${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$$denote the minimal element of the Collatz orbit$$N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $$. The infamousCollatz conjectureasserts that$${\operatorname {Col}}_{\min }(N)=1$$for all$$N \in \mathbb {N}+1$$. Previously, it was shown by Korec that for any$$\theta> \frac {\log 3}{\log 4} \approx 0.7924$$, one has$${\operatorname {Col}}_{\min }(N) \leq N^\theta $$for almost all$$N \in \mathbb {N}+1$$(in the sense of natural density). In this paper, we show that foranyfunction$$f \colon \mathbb {N}+1 \to \mathbb {R}$$with$$\lim _{N \to \infty } f(N)=+\infty $$, one has$${\operatorname {Col}}_{\min }(N) \leq f(N)$$for almost all$$N \in \mathbb {N}+1$$(in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a$$3$$-adic cyclic group$$\mathbb {Z}/3^n\mathbb {Z}$$at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency. 

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