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1. Estimates for smooth Weyl sums on minor arcs

   https://doi.org/10.1112/blms.13219  

   Brüdern, Jörg; Wooley, Trevor D (March 2025, Bulletin of the London Mathematical Society)

   We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of . In particular, when and is defined via the relation , then for all large numbers there is an integer with for which . 

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2. Small fractional parts of binary forms

   https://doi.org/10.1093/qmath/haad024  

   Yeon, Kiseok (June 2023, The Quarterly Journal of Mathematics)

   Abstract We obtain bounds on fractional parts of binary forms of the shape $$\Psi(x,y)=\alpha_k x^k+\alpha_l x^ly^{k-l}+\alpha_{l-1}x^{l-1}y^{k-l+1}+\cdots+\alpha_0 y^k$$ with $$\alpha_k,\alpha_l,\ldots,\alpha_0\in{\mathbb R}$$ and $$l\leq k-2.$$ By exploiting recent progress on Vinogradov’s mean value theorem and earlier work on exponential sums over smooth numbers, we derive estimates superior to those obtained hitherto for the best exponent σ, depending on k and $l,$ such that $$ \min_{\substack{0\leq x,y\leq X\\(x,y)\neq (0,0)}}\|\Psi(x,y)\|\leq X^{-\sigma+\epsilon}. $$ 

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3. Planar and poly-arc Lombardi drawings

   https://doi.org/https://doi.org/10.20382/jocg.v9i1a11  

   Duncan, Christian A.; Eppstein, David; Goodrich, Michael T.; Kobourov, Stephen G.; Löffler, Maarten; Nöllenburg, Martin (January 2018, Journal of computational geometry)

   In Lombardi drawings of graphs, edges are represented as circular arcs and the edges incident on vertices have perfect angular resolution. It is known that not every planar graph has a planar Lombardi drawing. We give an example of a planar 3-tree that has no planar Lombardi drawing and we show that all outerpaths do have a planar Lombardi drawing. Further, we show that there are graphs that do not even have any Lombardi drawing at all. With this in mind, we generalize the notion of Lombardi drawings to that of (smooth) k-Lombardi drawings, in which each edge may be drawn as a (differentiable) sequence of k circular arcs; we show that every graph has a smooth 2-Lombardi drawing and every planar graph has a smooth planar 3-Lombardi drawing. We further investigate related topics connecting planarity and Lombardi drawings. 

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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. Transport signatures of Fermi arcs at twin boundaries in Weyl materials

   https://doi.org/10.1103/PhysRevB.111.085133  

   Kaushik, Sahal; Robredo, Iñigo; Mathur, Nitish; Schoop, Leslie M; Jin, Song; Vergniory, Maia G; Cano, Jennifer (February 2025, Physical Review B)

   One of the most striking signatures of Weyl fermions in solid-state systems is their surface Fermi arcs. Fermi arcs can also be localized at internal twin boundaries where two Weyl materials of opposite chirality meet. In this work, we derive constraints on the topology and connectivity of these “internal Fermi arcs.” We show that internal Fermi arcs can exhibit transport signatures, and we propose two probes: quantum oscillations and a quantized chiral magnetic current. We propose merohedrally twinned B20 materials as candidates to host internal Fermi arcs, verified through both model and calculations. Our theoretical investigation sheds light on the topological features and motivates experimental studies on the intriguing physics of internal Fermi arcs. 

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