Near-Cramér Sato–Tate Equidistribution in Growing Arithmetic Progressions

## Overview Let \(E/\mathbf Q\) be a fixed non-CM elliptic curve, and let \(\theta_p\in[0,\pi]\) denote its Frobenius angle at a prime of good reduction \(p\). This paper studies Sato–Tate equidistribution for primes constrained simultaneously by a literal moving hard interval \[X 1\), take \[h=\sqrt X(\log X)^A.\] If \[\varphi(q)=o\!\left((\log X)^{A-1}\right),\] then, under the corresponding finite twisted-symmetric-power GRH packet, \[\sup_{I\subseteq[0,\pi]}\left|\mu^\sharp_{X,h;q,a}(I)-\mu_{\mathrm{ST}}(I)\right|\ll_{E,A}\left[\frac{\varphi(q)}{(\log X)^{A-1}}\left(1+\log\frac{(\log X)^{A-1}}{\varphi(q)}\right)\right]^{1/2}.\] For fixed \(q\), this recovers the near-square-root discrepancy rate \[\frac{\sqrt{\log\log X}}{(\log X)^{(A-1)/2}}.\] ## Position in the literature Effective Sato–Tate theorems in arithmetic progressions, Sato–Tate results in Hoheisel-type short intervals, and GRH Cramér-scale prime localization in arithmetic progressions are all known separately. The contribution here is the combination of a literal moving hard prime window near the arithmetic-progression Cramér regime, a growing modulus, uniform Sato–Tate angular discrepancy, and finite twisted-symmetric-power analytic depth. No unconditional priority claim is made for this precise combination. ## Version Version v1.0r1.