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1. Super-polynomial accuracy of one dimensional randomized nets using the median of means

   https://doi.org/10.1090/mcom/3791  

   Pan, Zexin; Owen, Art (March 2023, Mathematics of Computation)

   Let $f$be analytic on $[0,1]$with $|f^{\left(k\right)}\left(1/2\right)|⩽<\#comment/>A{\mathrm{\alpha <\#comment/>}}^{k}k!$for some constants $A$and $\mathrm{\alpha <\#comment/>}>2$and all $k⩾<\#comment/>1$. We show that the median estimate of $\mathrm{\mu <\#comment/>}={\int <\#comment/>}_0^{1}f\left(x\right)\mathrm{d}x$under random linear scrambling with $n=2^m$points converges at the rate $O\left(n^{-<\#comment/>c\log <\#comment/>\left(n\right)}\right)$for any $c>3\log <\#comment/>\left(2\right)/{\mathrm{\pi <\#comment/>}}^2\approx <\#comment/>0.21$. We also get a super-polynomial convergence rate for the sample median of $2k-<\#comment/>1$random linearly scrambled estimates, when $k/m$is bounded away from zero. When $f$has a $p$’th derivative that satisfies a $\mathrm{\lambda <\#comment/>}$-Hölder condition then the median of means has error $O\left(n^{-<\#comment/>(p+\mathrm{\lambda <\#comment/>})+\mathrm{ϵ<\#comment/>}}\right)$for any $\mathrm{ϵ<\#comment/>}>0$, if $k\to <\#comment/>\mathrm{\infty <\#comment/>}$as $m\to <\#comment/>\mathrm{\infty <\#comment/>}$. The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number. 

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2. A modular construction of unramified 𝑝-extensions of ℚ(ℕ^{1/𝕡})

   https://doi.org/10.1090/bproc/141  

   Lang, Jaclyn; Wake, Preston (October 2022, Proceedings of the American Mathematical Society, Series B)

   We show that for primes $N,p\ge <\#comment/>5$with $N\equiv <\#comment/>-<\#comment/>1modp$, the class number of $\mathbb{Q}\left(N^{1/p}\right)$is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N\equiv <\#comment/>-<\#comment/>1modp$, there is always a cusp form of weight $2$and level ${\mathrm{\Gamma <\#comment/>}}_0\left(N^2\right)$whose $\mathrm{\ell <\#comment/>}$th Fourier coefficient is congruent to $\mathrm{\ell <\#comment/>}+1$modulo a prime above $p$, for all primes $\mathrm{\ell <\#comment/>}$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree- $p$extension of $\mathbb{Q}\left(N^{1/p}\right)$. 

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3. On a conjectural symmetric version of Ehrhard’s inequality

   https://doi.org/10.1090/tran/9177  

   Livshyts, Galyna (May 2024, Transactions of the American Mathematical Society)

   We formulate a plausible conjecture for the optimal Ehrhard-type inequality for convex symmetric sets with respect to the Gaussian measure. Namely, letting $J_{k-<\#comment/>1}\left(s\right)={\int <\#comment/>}_0^s{t}^{k-<\#comment/>1}{e}^{-<\#comment/>\frac{t^2}{2}}dt$and $c_{k-<\#comment/>1}=J_{k-<\#comment/>1}(+\mathrm{\infty <\#comment/>})$, we conjecture that the function $F:[0,1]\to <\#comment/>\mathbb{R}$, given by $F\left(a\right)=\underset{k=1}{\overset{n}{\sum <\#comment/>}}{1}_{a\in <\#comment/>E_k}\cdot <\#comment/>\left({\mathrm{\beta <\#comment/>}}_k{J}_{k-<\#comment/>1}^{-<\#comment/>1}\right(c_{k-<\#comment/>1}a)+{\mathrm{\alpha <\#comment/>}}_k)$(with an appropriate choice of a decomposition $[0,1]={\cup <\#comment/>}_i{E}_i$and coefficients ${\mathrm{\alpha <\#comment/>}}_i,{\mathrm{\beta <\#comment/>}}_i$) satisfies, for all symmetric convex sets $K$and $L$, and any $\mathrm{\lambda <\#comment/>}\in <\#comment/>[0,1]$, $F\left(\mathrm{\gamma <\#comment/>}\right(\mathrm{\lambda <\#comment/>}K+(1-<\#comment/>\mathrm{\lambda <\#comment/>})L\left)\right)\ge <\#comment/>\mathrm{\lambda <\#comment/>}F\left(\mathrm{\gamma <\#comment/>}\right(K\left)\right)+(1-<\#comment/>\mathrm{\lambda <\#comment/>})F\left(\mathrm{\gamma <\#comment/>}\right(L\left)\right).$We explain that this conjecture is “the most optimistic possible”, and is equivalent to the fact that for any symmetric convex set $K$, itsGaussian concavity power $p_s(K,\mathrm{\gamma <\#comment/>})$is greater than or equal to $p_s(R{B}_2^k\times <\#comment/>{\mathbb{R}}^{n-<\#comment/>k},\mathrm{\gamma <\#comment/>})$, for some $k\in <\#comment/>\{1,\dots <\#comment/>,n\}$. We call the sets $R{B}_2^k\times <\#comment/>{\mathbb{R}}^{n-<\#comment/>k}$round $k$-cylinders; they also appear as the conjectured Gaussian isoperimetric minimizers for symmetric sets, see Heilman [Amer. J. Math. 143 (2021), pp. 53–94]. In this manuscript, we make progress towards this question, and show that for any symmetric convex set $K$in ${\mathbb{R}}^n$, $p_s(K,\mathrm{\gamma <\#comment/>})\ge <\#comment/>\underset{F\in <\#comment/>L^2(K,\mathrm{\gamma <\#comment/>})\cap <\#comment/>Lip\left(K\right):\int <\#comment/>F=1}{sup}\left(2T_{\mathrm{\gamma <\#comment/>}}^F\right(K)-<\#comment/>Var(F\left)\right)+\frac{1}{n-<\#comment/>\mathbb{E}{X}^2},$where $T_{\mathrm{\gamma <\#comment/>}}^F\left(K\right)$is the $F-<\#comment/>$torsional rigidity of $K$with respect to the Gaussian measure.Moreover, the equality holds if and only if $K=R{B}_2^k\times <\#comment/>{\mathbb{R}}^{n-<\#comment/>k}$for some $R>0$and $k=1,\dots <\#comment/>,n$.As a consequence, we get $p_s(K,\mathrm{\gamma <\#comment/>})\ge <\#comment/>Q(\mathbb{E}|X{|}^2,\mathbb{E}\Vert <\#comment/>X{\Vert <\#comment/>}_K^4,\mathbb{E}\Vert <\#comment/>X{\Vert <\#comment/>}_K^2,r(K\left)\right),$where $Q$is a certain rational function of degree $2$, the expectation is taken with respect to the restriction of the Gaussian measure onto $K$, $\Vert <\#comment/>\cdot <\#comment/>{\Vert <\#comment/>}_K$is the Minkowski functional of $K$, and $r\left(K\right)$is the in-radius of $K$. The result follows via a combination of some novel estimates, the $L2$method (previously studied by several authors, notably Kolesnikov and Milman [J. Geom. Anal. 27 (2017), pp. 1680–1702; Amer. J. Math. 140 (2018), pp. 1147–1185;Geometric aspects of functional analysis, Springer, Cham, 2017; Mem. Amer. Math. Soc. 277 (2022), v+78 pp.], Kolesnikov and the author [Adv. Math. 384 (2021), 23 pp.], Hosle, Kolesnikov, and the author [J. Geom. Anal. 31 (2021), pp. 5799–5836], Colesanti [Commun. Contemp. Math. 10 (2008), pp. 765–772], Colesanti, the author, and Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139], Eskenazis and Moschidis [J. Funct. Anal. 280 (2021), 19 pp.]), and the analysis of the Gaussian torsional rigidity. As an auxiliary result on the way to the equality case characterization, we characterize the equality cases in the “convex set version” of the Brascamp-Lieb inequality, and moreover, obtain a quantitative stability version in the case of the standard Gaussian measure; this may be of independent interest. All the equality case characterizations rely on the careful analysis of the smooth case, the stability versions via trace theory, and local approximation arguments. In addition, we provide a non-sharp estimate for a function $F$whose composition with $\mathrm{\gamma <\#comment/>}\left(K\right)$is concave in the Minkowski sense for all symmetric convex sets. 

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4. The best constant for 𝐿^{∞}-type Gagliardo-Nirenberg inequalities

   https://doi.org/10.1090/qam/1645  

   Liu, Jian-Guo; Wang, Jinhuan (June 2024, Quarterly of Applied Mathematics)

   In this paper we derive the best constant for the following $L^{\mathrm{\infty <\#comment/>}}$-type Gagliardo-Nirenberg interpolation inequality $\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{\mathrm{\infty <\#comment/>}}}\le <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{q+1}}^{1-<\#comment/>\mathrm{\theta <\#comment/>}}\Vert <\#comment/>\mathrm{\nabla <\#comment/>}u{\Vert <\#comment/>}_{L^p}^{\mathrm{\theta <\#comment/>}},\mathrm{\theta <\#comment/>}=\frac{pd}{dp+(p-<\#comment/>d)(q+1)},$where parameters $q$and $p$satisfy the conditions $p>d\ge <\#comment/>1$, $q\ge <\#comment/>0$. The best constant $C_{q,\mathrm{\infty <\#comment/>},p}$is given by $C_{q,\mathrm{\infty <\#comment/>},p}={\mathrm{\theta <\#comment/>}}^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{p}}(1-<\#comment/>\mathrm{\theta <\#comment/>}{)}^{\frac{\mathrm{\theta <\#comment/>}}{p}}{M}_c^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{d}},M_c≔{\int <\#comment/>}_{{\mathbb{R}}^d}{u}_{c,\mathrm{\infty <\#comment/>}}^{q+1}dx,$where $u_{c,\mathrm{\infty <\#comment/>}}$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when $u=A{u}_{c,\mathrm{\infty <\#comment/>}}\left(\mathrm{\lambda <\#comment/>}\right(x-<\#comment/>x_0\left)\right)$for any real numbers $A$, $\mathrm{\lambda <\#comment/>}>0$and $x_0\in <\#comment/>{\mathbb{R}}^d$. In fact, the generalized Lane-Emden equation in ${\mathbb{R}}^d$contains a delta function as a source and it is a Thomas-Fermi type equation. For $q=0$or $d=1$, $u_{c,\mathrm{\infty <\#comment/>}}$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that $u_{c,m}\to <\#comment/>u_{c,\mathrm{\infty <\#comment/>}}$and $C_{q,m,p}\to <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}$as $m\to <\#comment/>+\mathrm{\infty <\#comment/>}$for $d=1$, where $u_{c,m}$and $C_{q,m,p}$are the function achieving equality and the best constant of $L^m$-type Gagliardo-Nirenberg interpolation inequality, respectively. 

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5. Products of normal subsets

   https://doi.org/10.1090/tran/8960  

   Larsen, Michael; Shalev, Aner; Tiep, Pham (February 2024, Transactions of the American Mathematical Society)

   In this paper we consider which families of finite simple groups $G$have the property that for each $\mathrm{ϵ<\#comment/>}>0$there exists $N>0$such that, if $|G|\ge <\#comment/>N$and $S,T$are normal subsets of $G$with at least $\mathrm{ϵ<\#comment/>}|G|$elements each, then every non-trivial element of $G$is the product of an element of $S$and an element of $T$. We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form ${\mathrm{P}\mathrm{S}\mathrm{L}}_n\left(q\right)$where $q$is fixed and $n\to <\#comment/>\mathrm{\infty <\#comment/>}$. However, in the case $S=T$and $G$alternating this holds with an explicit bound on $N$in terms of $\mathrm{ϵ<\#comment/>}$. Related problems and applications are also discussed. In particular we show that, if $w_1,w_2$are non-trivial words, $G$is a finite simple group of Lie type of bounded rank, and for $g\in <\#comment/>G$, $P_{w_1\left(G\right),w_2\left(G\right)}\left(g\right)$denotes the probability that $g_1{g}_2=g$where $g_i\in <\#comment/>w_i\left(G\right)$are chosen uniformly and independently, then, as $|G|\to <\#comment/>\mathrm{\infty <\#comment/>}$, the distribution $P_{w_1\left(G\right),w_2\left(G\right)}$tends to the uniform distribution on $G$with respect to the $L^{\mathrm{\infty <\#comment/>}}$norm. 

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