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1. Averages Along the Primes: Improving and Sparse Bounds

   https://doi.org/10.1515/conop-2020-0003  

   Han, Rui; Krause, Ben; Lacey, Michael T.; Yang, Fan (February 2020, Concrete Operators)

   Abstract Consider averages along the prime integers ℙ given by {\mathcal{A}_N}f(x) = {N^{ - 1}}\sum\limits_{p \in \mathbb{P}:p \le N} {(\log p)f(x - p).} These averages satisfy a uniform scale-free ℓ p -improving estimate. For all 1 < p < 2, there is a constant C p so that for all integer N and functions f supported on [0, N ], there holds {N^{ - 1/p'}}{\left\| {{\mathcal{A}_N}f} \right\|_{\ell p'}} \le {C_p}{N^{ - 1/p}}{\left\| f \right\|_{\ell p}}. The maximal function 𝒜 * f = sup N |𝒜 N f | satisfies ( p , p ) sparse bounds for all 1 < p < 2. The latter are the natural variants of the scale-free bounds. As a corollary, 𝒜 * is bounded on ℓ p ( w ), for all weights w in the Muckenhoupt 𝒜 p class. No prior weighted inequalities for 𝒜 * were known. 

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2. Concatenating Bipartite Graphs

   https://doi.org/10.37236/8451  

   Chudnovsky, Maria; Hompe, Patrick; Scott, Alex; Seymour, Paul; Spirkl, Sophie (April 2022, The Electronic Journal of Combinatorics)

   Let $$x,y\in (0,1]$$, and let $A,B,C$ be disjoint nonempty stable subsets of a graph $$G$$, where every vertex in $$A$$ has at least $x|B|$ neighbours in $$B$$, and every vertex in $$B$$ has at least $y|C|$ neighbours in $$C$$, and there are no edges between $A,C$. We denote by $$\phi(x,y)$$ the maximum $$z$$ such that, in all such graphs $$G$$, there is a vertex $$v\in C$$ that is joined to at least $z|A|$ vertices in $$A$$ by two-edge paths. This function has some interesting properties: we show, for instance, that $$\phi(x,y)=\phi(y,x)$$ for all $x,y$, and there is a discontinuity in $$\phi(x,x)$$ when $1/x$ is an integer. For $z=1/2, 2/3,1/3,3/4,2/5,3/5$, we try to find the (complicated) boundary between the set of pairs $(x,y)$ with $$\phi(x,y)\ge z$$ and the pairs with $$\phi(x,y)1/3$. 

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3. Realizing trees of configurations in thin sets

   https://doi.org/10.2140/pjm.2025.335.355  

   Greenleaf, Allan; Iosevich, Alex; Taylor, Krystal (January 2025, Pacific Journal of Mathematics)

   Let $$\phi(x,y)$$ be a continuous function, smooth away from the diagonal, such that, for some $$\alpha>0$$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^{\phi}f(x)=\int_{\phi(x,y)=t} f(y) \psi(y) d\sigma_{x,t}(y) \end{equation} map $$L^2({\mathbb R}^d) \to H^{\alpha}({\mathbb R}^d)$$ for all $t>0$. Let $$E$$ be a compact subset of $${\mathbb R}^d$$ for some $$d \ge 2$$, and suppose that the Hausdorff dimension of $$E$$ is $$>d-\alpha$$. We show that any tree graph $$T$$ on $k+1$ ($$k \ge 1$$) vertices is realizable in $$E$$, in the sense that there exist distinct $$x^1, x^2, \dots, x^{k+1} \in E$$ and $t>0$ such that the $$\phi$$-distance $$\phi(x^i, x^j)$$ is equal to $$t$$ for all pairs $(i,j)$ corresponding to the edges of the graph $$T$$. 

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4. Weyl remainders: an application of geodesic beams

   https://doi.org/10.1007/s00222-023-01178-5  

   Canzani, Yaiza; Galkowski, Jeffrey (June 2023, Inventiones mathematicae)

   Abstract We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold ( M ,  g ) of dimension n , let $$\Pi _\lambda $$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(-\Delta _g)$$ 1 [ 0 , λ 2 ] ( - Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty $$ λ → ∞ $$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$ ∫ W Π λ ( x , x ) d x = ( 2 π ) - n vol R n ( B ) vol g ( W ) λ n + O ( λ n - 1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams. 

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5. Variation of canonical height for\break Fatou points on ℙ ¹

   https://doi.org/10.1515/crelle-2022-0078  

   DeMarco, Laura; Mavraki, Niki Myrto (December 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let f : ℙ 1 → ℙ 1 {f:\mathbb{P}^{1}\to\mathbb{P}^{1}} be a map of degree > 1 {>1} defined over a function field k = K ⁢ ( X ) {k=K(X)} , where K is a number field and X is a projective curve over K . For each point a ∈ ℙ 1 ⁢ ( k ) {a\in\mathbb{P}^{1}(k)} satisfying a dynamical stability condition, we prove that the Call–Silverman canonical height for specialization f t {f_{t}} at point a t {a_{t}} , for t ∈ X ⁢ ( ℚ ¯ ) {t\in X(\overline{\mathbb{Q}})} outside a finite set, induces a Weil height on the curve X ; i.e., we prove the existence of a ℚ {\mathbb{Q}} -divisor D = D f , a {D=D_{f,a}} on X so that the function t ↦ h ^ f t ⁢ ( a t ) - h D ⁢ ( t ) {t\mapsto\hat{h}_{f_{t}}(a_{t})-h_{D}(t)} is bounded on X ⁢ ( ℚ ¯ ) {X(\overline{\mathbb{Q}})} for any choice of Weil height associated to D . We also prove a local version, that the local canonical heights t ↦ λ ^ f t , v ⁢ ( a t ) {t\mapsto\hat{\lambda}_{f_{t},v}(a_{t})} differ from a Weil function for D by a continuous function on X ⁢ ( ℂ v ) {X(\mathbb{C}_{v})} , at each place v of the number field K . These results were known for polynomial maps f and all points a ∈ ℙ 1 ⁢ ( k ) {a\in\mathbb{P}^{1}(k)} without the stability hypothesis,[21, 14],and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a ∈ ℙ 1 ⁢ ( k ) {a\in\mathbb{P}^{1}(k)} . [32, 29].Finally, we characterize our stability condition in terms of the geometry of the induced map f ~ : X × ℙ 1 ⇢ X × ℙ 1 {\tilde{f}:X\times\mathbb{P}^{1}\dashrightarrow X\times\mathbb{P}^{1}} over K ; and we prove the existence of relative Néron models for the pair ( f , a ) {(f,a)} , when a is a Fatou point at a place γ of k , where the local canonical height λ ^ f , γ ⁢ ( a ) {\hat{\lambda}_{f,\gamma}(a)} can be computed as an intersection number. 

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