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1. Higher uniformity of arithmetic functions in short intervals I. All intervals

   https://doi.org/10.1017/fmp.2023.28  

   Matomäki, Kaisa; Shao, Xuancheng; Tao, Terence; Teräväinen, Joni (January 2023, Forum of Mathematics, Pi)

   Abstract We study higher uniformity properties of the Möbius function$$\mu $$, the von Mangoldt function$$\Lambda $$, and the divisor functions$$d_k$$on short intervals$$(X,X+H]$$with$$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$$for a fixed constant$$0 \leq \theta < 1$$and any$$\varepsilon>0$$. More precisely, letting$$\Lambda ^\sharp $$and$$d_k^\sharp $$be suitable approximants of$$\Lambda $$and$$d_k$$and$$\mu ^\sharp = 0$$, we show for instance that, for any nilsequence$$F(g(n)\Gamma )$$, we have$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when$$\theta = 5/8$$and$$f \in \{\Lambda , \mu , d_k\}$$or$$\theta = 1/3$$and$$f = d_2$$. As a consequence, we show that the short interval Gowers norms$$\|f-f^\sharp \|_{U^s(X,X+H]}$$are also asymptotically small for any fixedsfor these choices of$$f,\theta $$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$$L^2$$. Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type$$II$$sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$$I_2$$sums. 

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2. COVERING INTEGERS BY $x^2 + dy^2$

   https://doi.org/10.1017/S1474748024000513  

   Green, Ben; Soundararajan, Kannan (May 2025, Journal of the Institute of Mathematics of Jussieu)

   Abstract What proportion of integers$$n \leq N$$may be expressed as$$x^2 + dy^2$$for some$$d \leq \Delta $$, with$$x,y$$integers? Writing$$\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$$for some$$\alpha \in (-\infty , \infty )$$, we show that the answer is$$\Phi (\alpha ) + o(1)$$, where$$\Phi $$is the Gaussian distribution function$$\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$$. A consequence of this is a phase transition: Almost none of the integers$$n \leq N$$can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 - \varepsilon }$$, but almost all of them can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$$. 

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3. A virtual PGL–SL correspondence for projective surfaces

   https://doi.org/10.1112/mod.2024.13  

   van_Bree, Dirk; Gholampour, Amin; Jiang, Yunfeng; Kool, Martijn (January 2025, Moduli)

   Abstract For a smooth projective surface$$X$$satisfying$$H_1(X,\mathbb{Z}) = 0$$and$$w \in H^2(X,\mu _r)$$, we study deformation invariants of the pair$$(X,w)$$. Choosing a Brauer–Severi variety$$Y$$(or, equivalently, Azumaya algebra$$\mathcal{A}$$) over$$X$$with Stiefel–Whitney class$$w$$, the invariants are defined as virtual intersection numbers on suitable moduli spaces of stable twisted sheaves on$$Y$$constructed by Yoshioka (or, equivalently, moduli spaces of$$\mathcal{A}$$-modules of Hoffmann–Stuhler). We show that the invariants do not depend on the choice of$$Y$$. Using a result of de Jong, we observe that they are deformation invariants of the pair$$(X,w)$$. For surfaces with$$h^{2,0}(X) \gt 0$$, we show that the invariants can often be expressed as virtual intersection numbers on Gieseker–Maruyama–Simpson moduli spaces of stable sheaves on$$X$$. This can be seen as a$${\rm PGL}_r$$–$${\rm SL}_r$$correspondence. As an application, we express$${\rm SU}(r) / \mu _r$$Vafa–Witten invariants of$$X$$in terms of$${\rm SU}(r)$$Vafa–Witten invariants of$$X$$. We also show how formulae from Donaldson theory can be used to obtain upper bounds for the minimal second Chern class of Azumaya algebras on$$X$$with given division algebra at the generic point. 

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4. On approximability of satisfiable $$\boldsymbol {k}$$-CSPs: II

   https://doi.org/10.1017/S0963548325100114  

   Bhangale, Amey; Khot, Subhash; Minzer, Dor (November 2025, Combinatorics, Probability and Computing)

   Abstract Let$$\Sigma$$be an alphabet and$$\mu$$be a distribution on$$\Sigma ^k$$for some$$k \geqslant 2$$. Let$$\alpha \gt 0$$be the minimum probability of a tuple in the support of$$\mu$$(denoted$$\mathsf{supp}(\mu )$$). We treat the parameters$$\Sigma , k, \mu , \alpha$$as fixed and constant. We say that the distribution$$\mu$$has a linear embedding if there exist an Abelian group$$G$$(with the identity element$$0_G$$) and mappings$$\sigma _i : \Sigma \rightarrow G$$,$$1 \leqslant i \leqslant k$$, such that at least one of the mappings is non-constant and for every$$(a_1, a_2, \ldots , a_k)\in \mathsf{supp}(\mu )$$,$$\sum _{i=1}^k \sigma _i(a_i) = 0_G$$. In [Bhangale-Khot-Minzer, STOC 2022], the authors asked the following analytical question. Let$$f_i: \Sigma ^n\rightarrow [\!-1,1]$$be bounded functions, such that at least one of the functions$$f_i$$essentially has degree at least$$d$$, meaning that the Fourier mass of$$f_i$$on terms of degree less than$$d$$is at most$$\delta$$. If$$\mu$$has no linear embedding (over any Abelian group), then is it necessarily the case that\begin{equation*}\left | \mathop {\mathbb{E}}_{({\textbf {x}}_1, {\textbf {x}}_2, \ldots , {\textbf {x}}_k)\sim \mu ^{\otimes n}}[f_1({\textbf {x}}_1)f_2({\textbf {x}}_2)\cdots f_k({\textbf {x}}_k)] \right | = o_{d, \delta }(1),\end{equation*}where the right hand side$$\to 0$$as the degree$$d \to \infty$$and$$\delta \to 0$$? In this paper, we answer this analytical question fully and in the affirmative for$$k=3$$. We also show the following two applications of the result.1.The first application is related to hardness of approximation. Using the reduction from [5], we show that for every$$3$$-ary predicate$$P:\Sigma ^3 \to \{0,1\}$$such that$$P$$has no linear embedding, anSDP (semi-definite programming) integrality gap instanceof a$$P$$-Constraint Satisfaction Problem (CSP) instance with gap$$(1,s)$$can be translated into a dictatorship test with completeness$$1$$and soundness$$s+o(1)$$, under certain additional conditions on the instance.2.The second application is related to additive combinatorics. We show that if the distribution$$\mu$$on$$\Sigma ^3$$has no linear embedding, marginals of$$\mu$$are uniform on$$\Sigma$$, and$$(a,a,a)\in \texttt{supp}(\mu )$$for every$$a\in \Sigma$$, then every large enough subset of$$\Sigma ^n$$contains a triple$$({\textbf {x}}_1, {\textbf {x}}_2,{\textbf {x}}_3)$$from$$\mu ^{\otimes n}$$(and in fact a significant density of such triples). 

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5. On cohesive powers of linear orders

   https://doi.org/10.1017/jsl.2023.14  

   Dimitrov, Rumen; Harizanov, Valentina; Morozov, Andrey; Shafer, Paul; Soskova, Alexandra A; Vatev, Stefan V (September 2023, The Journal of Symbolic Logic)

   Abstract Cohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let$$\omega $$,$$\zeta $$, and$$\eta $$denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of$$\omega $$. If$$\mathcal {L}$$is a computable copy of$$\omega $$that is computably isomorphic to the usual presentation of$$\omega $$, then every cohesive power of$$\mathcal {L}$$has order-type$$\omega + \zeta \eta $$. However, there are computable copies of$$\omega $$, necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to$$\omega + \zeta \eta $$. For example, we show that there is a computable copy of$$\omega $$with a cohesive power of order-type$$\omega + \eta $$. Our most general result is that if$$X \subseteq \mathbb {N} \setminus \{0\}$$is a Boolean combination of$$\Sigma _2$$sets, thought of as a set of finite order-types, then there is a computable copy of$$\omega $$with a cohesive power of order-type$$\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$$, where$$\boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$$denotes the shuffle of the order-types inXand the order-type$$\omega + \zeta \eta + \omega ^*$$. Furthermore, ifXis finite and non-empty, then there is a computable copy of$$\omega $$with a cohesive power of order-type$$\omega + \boldsymbol {\sigma }(X)$$. 

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