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1. An almost linear time algorithm testing whether the Markoff graph modulo p is connected

   https://doi.org/10.1007/s40993-024-00592-9  

   Brown, Colby Austin (March 2025, Research in Number Theory)

   Abstract The Markoff graphs modulopwere proven by Chen (Ann Math 199(1), 2024) to be connected for all but finitely many primes, and Baragar (The Markoff equation and equations of Hurwitz. Brown University, 1991) conjectured that they are connected for all primes, equivalently that every solution to the Markoff equation moduloplifts to a solution over$$\mathbb {Z}$$ $Z$. In this paper, we provide an algorithmic realization of the process introduced by Bourgain et al. [arXiv:1607.01530] to test whether the Markoff graph modulopis connected for arbitrary primes. Our algorithm runs in$$o(p^{1 + \epsilon })$$ $o\left(p^{1+ϵ}\right)$time for every$$\epsilon > 0$$ $ϵ>0$. We demonstrate this algorithm by confirming that the Markoff graph modulopis connected for all primes less than one million. 

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2. Products of primes in arithmetic progressions

   https://doi.org/10.1515/crelle-2023-0096  

   Matomäki, Kaisa; Teräväinen, Joni (January 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract A conjecture of Erdős states that, for any large primeq, every reduced residue class $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be represented as a product $p_1{p}_2${p_{1}p_{2}}of two primes $p_1,p_2\le q${p_{1},p_{2}\leq q}. We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integerq, every reduced residue class $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be written as $p_1{p}_2{p}_3${p_{1}p_{2}p_{3}}with $p_1,p_2,p_3\le q${p_{1},p_{2},p_{3}\leq q}primes. We also show that, for any $\epsilon >0${\varepsilon>0}and any sufficiently large integerq, at least $\left(\frac{2}{3}-\epsilon \right)\phi \left(q\right)${(\frac{2}{3}-\varepsilon)\varphi(q)}reduced residue classes $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be represented as a product $p_1{p}_2${p_{1}p_{2}}of two primes $p_1,p_2\le q${p_{1},p_{2}\leq q}.The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we establish bounds for the logarithmic density of primes in certain unions of cosets of subgroups of ${\mathbb{Z}}_q^{\times }${\mathbb{Z}_{q}^{\times}}of small index and study in detail the exceptional case that there exists a quadratic character $\psi \left(\mathrm{mod}q\right)${\psi~{}(\operatorname{mod}\,q)}such that $\psi \left(p\right)=-1${\psi(p)=-1}for very many primes $p\le q${p\leq q}. 

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3. Towards a Schinzel–Wójcik theorem for number fields

   https://doi.org/10.1007/s40879-025-00819-8  

   Pollack, Paul (April 2025, European Journal of Mathematics)

   Abstract Schinzel and Wójcik have shown that for every$$\alpha ,\beta \in \mathbb {Q}^{\times }\hspace{0.55542pt}{\setminus }\hspace{1.111pt}\{\pm 1\}$$ $\alpha ,\beta \in Q^{\times }\backslash \{\pm 1\}$, there are infinitely many primespwhere$$v_p(\alpha )=v_p(\beta )=0$$ $v_p\left(\alpha \right)=v_p\left(\beta \right)=0$and where$$\alpha $$ $\alpha$and$$\beta $$ $\beta$generate the same multiplicative group modp. We prove a weaker result in the same direction for algebraic numbers$$\alpha , \beta $$ $\alpha ,\beta$. Let$$\alpha , \beta \in \overline{\mathbb {Q}} ^{\times }$$ $\alpha ,\beta \in {\overline{Q}}^{\times }$, and suppose$$|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\alpha )|\ne 1$$ $|N_{Q(\alpha ,\beta )/Q}\left(\alpha \right)|\ne 1$and$$|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\beta )|\ne 1$$ $|N_{Q(\alpha ,\beta )/Q}\left(\beta \right)|\ne 1$. Then for some positive integer$$C = C(\alpha ,\beta )$$ $C=C(\alpha ,\beta )$, there are infinitely many prime idealsPof Equation missing<#comment/>where$$v_P(\alpha )=v_P(\beta )=0$$ $v_P\left(\alpha \right)=v_P\left(\beta \right)=0$and where the group$$\langle \beta \bmod {P}\rangle $$ $⟨\beta modP⟩$is a subgroup of$$\langle \alpha \bmod {P}\rangle $$ $⟨\alpha modP⟩$with$$[\langle \alpha \bmod {P}\rangle \,{:}\, \langle \beta \bmod {P}\rangle ]$$ $[⟨\alpha modP⟩:⟨\beta modP⟩]$dividingC. A key component of the proof is a theorem of Corvaja and Zannier bounding the greatest common divisor of shiftedS-units. 

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4. Regularity of area minimizing currents mod p

   https://doi.org/10.1007/s00039-020-00546-0  

   De Lellis, Camillo; Hirsch, Jonas; Marchese, Andrea; Stuvard, Salvatore (October 2020, Geometric and Functional Analysis)

   Abstract We establish a first general partial regularity theorem for area minimizing currents$${\mathrm{mod}}(p)$$ $\mathrm{mod}\left(p\right)$, for everyp, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of anm-dimensional area minimizing current$${\mathrm{mod}}(p)$$ $\mathrm{mod}\left(p\right)$cannot be larger than$$m-1$$ $m-1$. Additionally, we show that, whenpis odd, the interior singular set is$$(m-1)$$ $(m-1)$-rectifiable with locally finite$$(m-1)$$ $(m-1)$-dimensional measure. 

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5. On the Moduli of Lipschitz Homology Classes

   https://doi.org/10.1007/s12220-025-01916-6  

   Kangasniemi, Ilmari; Prywes, Eden (February 2025, The Journal of Geometric Analysis)

   Abstract We define a type of modulus$$\operatorname {dMod}_p$$ ${dMod}_p$for Lipschitz surfaces based on$$L^p$$ $L^p$-integrable measurable differential forms, generalizing the vector modulus of Aikawa and Ohtsuka. We show that this modulus satisfies a homological duality theorem, where for Hölder conjugate exponents$$p, q \in (1, \infty )$$ $p,q\in (1,\infty )$, every relative Lipschitzk-homology classchas a unique dual Lipschitz$$(n-k)$$ $(n-k)$-homology class$$c'$$ $c^{\prime }$such that$$\operatorname {dMod}_p^{1/p}(c) \operatorname {dMod}_q^{1/q}(c') = 1$$ ${dMod}_p^{1/p}\left(c\right){dMod}_q^{1/q}\left(c^{\prime }\right)=1$and the Poincaré dual ofcmaps$$c'$$ $c^{\prime }$to 1. As$$\operatorname {dMod}_p$$ ${dMod}_p$is larger than the classical surface modulus$$\operatorname {Mod}_p$$ ${Mod}_p$, we immediately recover a more general version of the estimate$$\operatorname {Mod}_p^{1/p}(c) \operatorname {Mod}_q^{1/q}(c') \le 1$$ ${Mod}_p^{1/p}\left(c\right){Mod}_q^{1/q}\left(c^{\prime }\right)\le 1$, which appears in works by Freedman and He and by Lohvansuu. Our theory is formulated in the general setting of Lipschitz Riemannian manifolds, though our results appear new in the smooth setting as well. We also provide a characterization of closed and exact Sobolev forms on Lipschitz manifolds based on integration over Lipschitzk-chains. 

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