Validity of approximation method for von Mangoldt function

Question

I'm working on a problem involving the pointwise almost everywhere convergence of multilinear ergodic averages with the von Mangoldt function inspired by this paper. Specifically, I'm looking at averages of the form:

$$ A_N(x) = \mathbb{E}_{n \in [1,N]} \Lambda(n) f_1(T^{P_1(n)}x) f_2(T^{P_2(n)}x) \ldots f_k(T^{P_k(n)}x) $$

where $\Lambda(n)$ is the von Mangoldt function, $T$ is a measure-preserving transformation, and $P_i(n)$ are polynomials.

After considering various approaches, I've developed a hybrid method to approximate $\Lambda(n)$. At a high level, this what I've tried:

1. Truncate $\Lambda_M(n)$ $$ \Lambda_M(n) = \begin{cases} \Lambda(n) & \text{if } n \leq M, \\ 0 & \text{otherwise}. \end{cases} $$ where $M = N^\theta$ for some $0 < \theta < 1$.

2. Decompose $\Lambda_M(n)$ into structured and pseudorandom parts $$ \Lambda_M(n) = \Lambda_{\text{str}}(n) + \Lambda_{\text{rand}}(n) $$

3. For the structured part, I've tried $$ \Lambda_{\text{str}}(n) = \sum_{p \leq M^{1/2}} \log p \cdot \mathbf{1}_{p | n} $$

4. Ensure that the pseudorandom part satisfies $$ \|\Lambda_{\text{rand}}\|_{U^{d+1}[N]} \leq \epsilon(N) $$ where $U^{d+1}[N]$ is the Gowers uniformity norm.

5. To further refine this, I've considered a Fourier-based approach for $\Lambda_{\text{str}}(n)$: $$ \Lambda_{\text{str}}(n) = \sum_{|\xi| \leq K} \hat{\Lambda}(\xi) e^{2\pi i \xi n / N} $$ where $K$ is chosen such that $\sum_{|\xi| > K} |\hat{\Lambda}(\xi)|^2 \leq \epsilon(N)$.

6. I'm trying to control the overall approximation error by ensuring $$ \|\Lambda - \Lambda_M\|_{U^{d+1}[N]} \leq \delta(N) $$ where $\delta(N) \to 0$ as $N \to \infty$.

Is this a valid approximation method for $\Lambda(n)$ in the context of multilinear ergodic averages? Are there any potential issues or improvements I should consider? Any insights or suggestions would be greatly appreciated. Thank you!

Answer 1

I accidentally logged out of my account and can't comment. To answer Yemon Choi's comment: An effective approximation splits $\Lambda(n)$ into a structured part that preserves essential arithmetic correlations and a pseudorandom part whose Gowers $U^{d+1}[N]$ norm becomes negligible as $N$ grows. Specifically, decomposing $$\Lambda(n) = \Lambda_{\text{str}}(n) + \Lambda_{\text{rand}}(n)$$ where $\Lambda_{\text{rand}}$ has small $U^{d+1}[N]$ norm ensures that the multilinear averages $A_N(x)$ are governed by the structured component, which ensures almost everywhere convergence.

The Cramér approximant mentioned in the paper has good control in the $U^3[N]$ norm (for the bilinear case with quadratic polynomials). However, for the multilinear case with polynomials of degree $d$, we need control in the $U^{d+1}[N]$ norm. It's not clear whether the Cramér approximant would have sufficiently rapid decay in these higher-order Gowers norms which is what my proposed approximation intends to do.