Pointwise convergence of bilinear polynomial averages over the primes
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| Format: | Preprint |
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2024
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| author | Krause, Ben Mousavi, Hamed Tao, Terence Teräväinen, Joni |
| author_facet | Krause, Ben Mousavi, Hamed Tao, Terence Teräväinen, Joni |
| contents | We show that on a $σ$-finite measure preserving system $X = (X,ν, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} Λ(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$, $g \in L^{p_2}(X)$, and $1/p_1 + 1/p_2 \leq 1$, where $P$ is a polynomial with integer coefficients of degree at least $2$. This had previously been established with the von Mangoldt weight $Λ$ replaced by the constant weight $1$ by the first and third authors with Mirek, and by the Möbius weight $μ$ by the fourth author. The proof is based on combining tools from both of these papers, together with several Gowers norm and polynomial averaging operator estimates on approximants to the von Mangoldt function of ''Cramér'' and ''Heath-Brown'' type. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_10510 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pointwise convergence of bilinear polynomial averages over the primes Krause, Ben Mousavi, Hamed Tao, Terence Teräväinen, Joni Dynamical Systems Classical Analysis and ODEs Number Theory 37A30, 37A44, 37A46, 11B30 We show that on a $σ$-finite measure preserving system $X = (X,ν, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} Λ(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$, $g \in L^{p_2}(X)$, and $1/p_1 + 1/p_2 \leq 1$, where $P$ is a polynomial with integer coefficients of degree at least $2$. This had previously been established with the von Mangoldt weight $Λ$ replaced by the constant weight $1$ by the first and third authors with Mirek, and by the Möbius weight $μ$ by the fourth author. The proof is based on combining tools from both of these papers, together with several Gowers norm and polynomial averaging operator estimates on approximants to the von Mangoldt function of ''Cramér'' and ''Heath-Brown'' type. |
| title | Pointwise convergence of bilinear polynomial averages over the primes |
| topic | Dynamical Systems Classical Analysis and ODEs Number Theory 37A30, 37A44, 37A46, 11B30 |
| url | https://arxiv.org/abs/2409.10510 |