A Variation Norm Carleson Theorem Along the Primes

Let $\Lambda$ denote the von Mangoldt function; we prove that for each $r > 2$, there exist constants \[ r' < \mathbf{c}(r) < 2 < \mathbf{C}(r), \qquad \lim_{r \to \infty} \mathbf{c}(r) = 1, \ \lim_{r \to \infty} \mathbf{C}(r) = \infty \] so that the discrete variational Carleson operator along the primes \begin{align} \mathcal{V}^r \Big( \sum_{n \neq 0} f(x-n) \Lambda(|n|) \frac{e^{2\pi i \lambda n}}{n} : \lambda \in \mathbb{T} \Big) \end{align} is bounded on $\ell^p$ for all $\mathbf{c}(r) < p < \mathbf{C}(r)$, while the variation is unbounded when $p \leq r'$. At the non-variational endpoint, the same argument gives the sharp maximal result: the prime Carleson operator \[ \sup_{\lambda\in\mathbb T} \Big|\sum_{n\neq0} f(x-n)\Lambda(|n|)\frac{e^{2\pi i\lambda n}}{n}\Big| \] is bounded on \(\ell^p(\mathbb Z)\) for the full expected range \(1<p<\infty\).
The proof gives a new mechanism for treating modulation-invariant singular integrals after arithmetic sparsification. It combines higher-order Fourier uniformity, a variable-coefficient multi-frequency principle in the spirit of Bourgain, and an additive-combinatorial inverse argument. A key step is a reduction to finite periodic models, where the Ramanujan structure of the major arcs is converted into a sharp estimate for structured atoms by elementary number theory.