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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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Higher uniformity of arithmetic functions in short intervals I. All intervals
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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- PAR ID:
- 10517138
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Forum of Mathematics, Pi
- Volume:
- 11
- ISSN:
- 2050-5086
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fmp.2023.28
