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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. 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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3. Improved effective Łojasiewicz inequality and applications

   https://doi.org/10.1017/fms.2024.66  

   Basu, Saugata; Mohammad-Nezhad, Ali (December 2024, Forum of Mathematics, Sigma)

   Abstract Let$$\mathrm {R}$$be a real closed field. Given a closed and bounded semialgebraic set$$A \subset \mathrm {R}^n$$and semialgebraic continuous functions$$f,g:A \rightarrow \mathrm {R}$$such that$$f^{-1}(0) \subset g^{-1}(0)$$, there exist an integer$$N> 0$$and$$c \in \mathrm {R}$$such that the inequality (Łojasiewicz inequality)$$|g(x)|^N \le c \cdot |f(x)|$$holds for all$$x \in A$$. In this paper, we consider the case whenAis defined by a quantifier-free formula with atoms of the form$$P = 0, P>0, P \in \mathcal {P}$$for some finite subset of polynomials$$\mathcal {P} \subset \mathrm {R}[X_1,\ldots ,X_n]_{\leq d}$$, and the graphs of$$f,g$$are also defined by quantifier-free formulas with atoms of the form$$Q = 0, Q>0, Q \in \mathcal {Q}$$, for some finite set$$\mathcal {Q} \subset \mathrm {R}[X_1,\ldots ,X_n,Y]_{\leq d}$$. We prove that the Łojasiewicz exponent in this case is bounded by$$(8 d)^{2(n+7)}$$. Our bound depends ondandnbut is independent of the combinatorial parameters, namely the cardinalities of$$\mathcal {P}$$and$$\mathcal {Q}$$. The previous best-known upper bound in this generality appeared inP. Solernó, Effective Łojasiewicz Inequalities in Semi-Algebraic Geometry, Applicable Algebra in Engineering, Communication and Computing (1991)and depended on the sum of degrees of the polynomials defining$$A,f,g$$and thus implicitly on the cardinalities of$$\mathcal {P}$$and$$\mathcal {Q}$$. As a consequence, we improve the current best error bounds for polynomial systems under some conditions. Finally, we prove a version of Łojasiewicz inequality in polynomially bounded o-minimal structures. We prove the existence of a common upper bound on the Łojasiewicz exponent for certain combinatorially defined infinite (but not necessarily definable) families of pairs of functions. This improves a prior result of Chris Miller (C. Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic (1994)). 

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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. Near-contact approach of two permeable spheres

   https://doi.org/10.1017/jfm.2021.588  

   Reboucas, Rodrigo B; Loewenberg, Michael (October 2021, Journal of Fluid Mechanics)

   An analysis is presented for the axisymmetric lubrication resistance between permeable spherical particles. Darcy's law is used to describe the flow in the permeable medium and a slip boundary condition is applied at the interface. The pressure in the near-contact region is governed by a non-local integral equation. The asymptotic limit$$K=k/a^{2} \ll 1$$is considered, where$$k$$is the arithmetic mean permeability, and$$a^{-1}=a^{-1}_{1}+a^{-1}_{2}$$is the reduced radius, and$$a_1$$and$$a_2$$are the particle radii. The formulation allows for particles with distinct particle radii, permeabilities and slip coefficients, including permeable and impermeable particles and spherical drops. Non-zero particle permeability qualitatively affects the axisymmetric near-contact motion, removing the classical lubrication singularity for impermeable particles, resulting in finite contact times under the action of a constant force. The lubrication resistance becomes independent of gap and attains a maximum value at contact$$F=6{\rm \pi} \mu a W K^{-2/5}\tilde {f}_c$$, where$$\mu$$is the fluid viscosity,$$W$$is the relative velocity and$$\tilde {f}_c$$depends on slip coefficients and weakly on permeabilities; for two permeable particles with no-slip boundary conditions,$$\tilde {f}_c=0.7507$$; for a permeable particle in near contact with a spherical drop,$$\tilde {f}_c$$is reduced by a factor of$$2^{-6/5}$$. 

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