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1. PETERZIL–STEINHORN SUBGROUPS AND -STABILIZERS IN ACF

   https://doi.org/10.1017/S147474802100030X  

   Kamensky, Moshe; Starchenko, Sergei; Ye, Jinhe (May 2023, Journal of the Institute of Mathematics of Jussieu)

   Abstract We considerG, a linear algebraic group defined over$$\Bbbk $$, an algebraically closed field (ACF). By considering$$\Bbbk $$as an embedded residue field of an algebraically closed valued fieldK, we can associate to it a compactG-space$$S^\mu _G(\Bbbk )$$consisting of$$\mu $$-types onG. We show that for each$$p_\mu \in S^\mu _G(\Bbbk )$$,$$\mathrm {Stab}^\mu (p)=\mathrm {Stab}\left (p_\mu \right )$$is a solvable infinite algebraic group when$$p_\mu $$is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of$$\mathrm {Stab}\left (p_\mu \right )$$in terms of the dimension ofp. 

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2. Almost Everywhere Behavior of Functions According to Partition Measures

   https://doi.org/10.1017/fms.2023.130  

   Chan, William; Jackson, Stephen; Trang, Nam (January 2024, Forum of Mathematics, Sigma)

   Abstract This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals. The following summarizes the main results proved under suitable partition hypotheses.•If$$\kappa $$is a cardinal,$$\epsilon < \kappa $$,$${\mathrm {cof}}(\epsilon ) = \omega $$,$$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$$and$$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$$, then$$\Phi $$satisfies the almost everywhere short length continuity property: There is a club$$C \subseteq \kappa $$and a$$\delta < \epsilon $$so that for all$$f,g \in [C]^\epsilon _*$$, if$$f \upharpoonright \delta = g \upharpoonright \delta $$and$$\sup (f) = \sup (g)$$, then$$\Phi (f) = \Phi (g)$$.•If$$\kappa $$is a cardinal,$$\epsilon $$is countable,$$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$$holds and$$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$$, then$$\Phi $$satisfies the strong almost everywhere short length continuity property: There is a club$$C \subseteq \kappa $$and finitely many ordinals$$\delta _0, ..., \delta _k \leq \epsilon $$so that for all$$f,g \in [C]^\epsilon _*$$, if for all$$0 \leq i \leq k$$,$$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$$, then$$\Phi (f) = \Phi (g)$$.•If$$\kappa $$satisfies$$\kappa \rightarrow _* (\kappa )^\kappa _2$$,$$\epsilon \leq \kappa $$and$$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$$, then$$\Phi $$satisfies the almost everywhere monotonicity property: There is a club$$C \subseteq \kappa $$so that for all$$f,g \in [C]^\epsilon _*$$, if for all$$\alpha < \epsilon $$,$$f(\alpha ) \leq g(\alpha )$$, then$$\Phi (f) \leq \Phi (g)$$.•Suppose dependent choice ($$\mathsf {DC}$$),$${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$$and the almost everywhere short length club uniformization principle for$${\omega _1}$$hold. Then every function$$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$$satisfies a finite continuity property with respect to closure points: Let$$\mathfrak {C}_f$$be the club of$$\alpha < {\omega _1}$$so that$$\sup (f \upharpoonright \alpha ) = \alpha $$. There is a club$$C \subseteq {\omega _1}$$and finitely many functions$$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$$so that for all$$f \in [C]^{\omega _1}_*$$, for all$$g \in [C]^{\omega _1}_*$$, if$$\mathfrak {C}_g = \mathfrak {C}_f$$and for all$$i < n$$,$$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$$, then$$\Phi (g) = \Phi (f)$$.•Suppose$$\kappa $$satisfies$$\kappa \rightarrow _* (\kappa )^\epsilon _2$$for all$$\epsilon < \kappa $$. For all$$\chi < \kappa $$,$$[\kappa ]^{<\kappa }$$does not inject into$${}^\chi \mathrm {ON}$$, the class of$$\chi $$-length sequences of ordinals, and therefore,$$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$$. As a consequence, under the axiom of determinacy$$(\mathsf {AD})$$, these two cardinality results hold when$$\kappa $$is one of the following weak or strong partition cardinals of determinacy:$${\omega _1}$$,$$\omega _2$$,$$\boldsymbol {\delta }_n^1$$(for all$$1 \leq n < \omega $$) and$$\boldsymbol {\delta }^2_1$$(assuming in addition$$\mathsf {DC}_{\mathbb {R}}$$). 

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3. 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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4. Multi-linear forms, graphs, and $L^p$-improving measures in $${\Bbb F}_q^d$$

   https://doi.org/10.4153/S0008414X2300086X  

   Bhowmik, Pablo; Iosevich, Alex; Koh, Doowon; Pham, Thang (February 2025, Canadian journal of mathematics)

   Abstract The purpose of this paper is to introduce and study the following graph-theoretic paradigm. Let$$ \begin{align*}T_Kf(x)=\int K(x,y) f(y) d\mu(y),\end{align*} $$where$$f: X \to {\Bbb R}$$,Xa set, finite or infinite, andKand$$\mu $$denote a suitable kernel and a measure, respectively. Given a connected ordered graphGonnvertices, consider the multi-linear form$$ \begin{align*}\Lambda_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) d\mu(x^l),\end{align*} $$where$${\mathcal E}(G)$$is the edge set ofG. Define$$\Lambda _G(p_1, \ldots , p_n)$$as the smallest constant$$C>0$$such that the inequality(0.1)$$ \begin{align} \Lambda_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, \mu)} \end{align} $$holds for all nonnegative real-valued functions$$f_i$$,$$1\le i\le n$$, onX. The basic question is, how does the structure ofGand the mapping properties of the operator$$T_K$$influence the sharp exponents in (0.1). In this paper, this question is investigated mainly in the case$$X={\Bbb F}_q^d$$, thed-dimensional vector space over the field withqelements,$$K(x^i,x^j)$$is the indicator function of the sphere evaluated at$$x^i-x^j$$, and connected graphsGwith at most four vertices. 

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5. 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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