Forum of Mathematics Pi最新文献_第3页

The Local Langlands Conjecture for 的局部朗兰兹猜想

1区数学

Forum of Mathematics Pi Pub Date : 2023-01-01 DOI: 10.1017/fmp.2023.27

Wee Teck Gan, Gordan Savin

引用次数: 10

Higher uniformity of arithmetic functions in short intervals I. All intervals 算术函数在短区间内的一致性更高

1区数学

Forum of Mathematics Pi Pub Date : 2023-01-01 DOI: 10.1017/fmp.2023.28

Kaisa Matomäki, Xuancheng Shao, Terence Tao, Joni Teräväinen

{"title":"Higher uniformity of arithmetic functions in short intervals I. All intervals","authors":"Kaisa Matomäki, Xuancheng Shao, Terence Tao, Joni Teräväinen","doi":"10.1017/fmp.2023.28","DOIUrl":"https://doi.org/10.1017/fmp.2023.28","url":null,"abstract":"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 fixed s for 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.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135053473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 6

On Thakur’s basis conjecture for multiple zeta values in positive characteristic 正特征中多个zeta值的塔库尔基猜想

1区数学

Forum of Mathematics Pi Pub Date : 2023-01-01 DOI: 10.1017/fmp.2023.26

Chieh-Yu Chang, Yen-Tsung Chen, Yoshinori Mishiba

引用次数: 3

Skew RSK dynamics: Greene invariants, affine crystals and applications to q-Whittaker polynomials 偏态RSK动力学:格林不变量、仿射晶体及其在q-Whittaker多项式中的应用

1区数学

Forum of Mathematics Pi Pub Date : 2023-01-01 DOI: 10.1017/fmp.2023.23

Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto

{"title":"Skew RSK dynamics: Greene invariants, affine crystals and applications to <i>q</i>-Whittaker polynomials","authors":"Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto","doi":"10.1017/fmp.2023.23","DOIUrl":"https://doi.org/10.1017/fmp.2023.23","url":null,"abstract":"Abstract Iterating the skew RSK correspondence discovered by Sagan and Stanley in the late 1980s, we define deterministic dynamics on the space of pairs of skew Young tableaux $(P,Q)$ . We find that these skew RSK dynamics display conservation laws which, in the picture of Viennot’s shadow line construction, identify generalizations of Greene invariants. The introduction of a novel realization of $0$ -th Kashiwara operators reveals that the skew RSK dynamics possess symmetries induced by an affine bicrystal structure, which, combined with connectedness properties of Demazure crystals, leads to the linearization of the time evolution. Studying asymptotic evolution of the dynamics started from a pair of skew tableaux $(P,Q)$ , we discover a new bijection $Upsilon : (P,Q) mapsto (V,W; kappa , nu )$ . Here, $(V,W)$ is a pair of vertically strict tableaux, that is, column strict fillings of Young diagrams with no condition on rows, with the shape prescribed by the Greene invariant, $kappa $ is an array of nonnegative weights and $nu $ is a partition. An application of this construction is the first bijective proof of Cauchy and Littlewood identities involving q -Whittaker polynomials. New identities relating sums of q -Whittaker and Schur polynomials are also presented.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 10

On a multi-parameter variant of the Bellow–Furstenberg problem 关于Bellow-Furstenberg问题的一个多参数变体

1区数学

Forum of Mathematics Pi Pub Date : 2023-01-01 DOI: 10.1017/fmp.2023.21

Jean Bourgain, Mariusz Mirek, Elias M. Stein, James Wright

引用次数: 8

Stellahedral geometry of matroids 拟阵的星面体几何

1区数学

Forum of Mathematics Pi Pub Date : 2023-01-01 DOI: 10.1017/fmp.2023.24

Christopher Eur, June Huh, Matt Larson

引用次数: 18

Interpolation for Brill–Noether curves Brill-Noether曲线的插值