Conditional Sato–Tate Localization in Growing Arithmetic Progressions: Critical Cramér-Order Occupancy and Dimension-Dependent Equidistribution

Abstract We study the Sato–Tate distribution when primes are constrained simultaneously to the moving interval $(X, X+h]$ and a single arithmetic progression (AP) $a \pmod q$ whose modulus $q = q(X)$ may grow with $X$. For a fixed elliptic curve without complex multiplication (non-CM), and for a pair of fixed twist-inequivalent non-CM elliptic curves, we prove conditional localization in unsmoothed prime intervals under the Generalized Riemann Hypothesis (GRH) for the relevant Dirichlet-twisted symmetric-power and bidegree Rankin–Selberg packets. A two-scale plateau explicit formula, followed by Dirichlet recombination before prime-power removal and passage to the sharp interval, yields at harmonic depth $M$ the interval and rectangle discrepancy bounds $$D^{(d)} \ll M^{-1} + M^d \mathcal{R}_{M,q}^{(d),\mathrm{AP}}(X, h, \Delta), \qquad d = 1, 2.$$ The one-variable estimates use only the coordinate-axis $L$-functions of the joint family and require $M$ angular modes; the joint rectangle estimate accumulates $M^2$ bidegrees. At the balanced choice $\Delta = \varphi(q)\sqrt{X}\log X$, fixed truncation depth $M$ already gives occupancy of the expected order: for every fixed positive-measure interval or rectangle, the corresponding prime count is $$\asymp \frac{h}{\varphi(q)\log X}$$ once $h \ge G\varphi(q)\sqrt{X}\log X$ with $G$ sufficiently large, and is $\asymp \sqrt{X}$ when $G$ is a fixed admissible constant. With growing depth, writing $\eta = (1 + \log G)/G + h/(X\log X)$, optimization gives the dimension-dependent rates $D^{(1)} \ll \eta^{1/2}$ and $D^{(2)} \ll \eta^{1/3}$. In the balanced Cramér-scale results, the frequently imposed auxiliary restriction $q \le X^{1/4}$ is unnecessary: the natural interval-length condition already implies $q \ll \sqrt{X}\log\log(3X)/\log X$. Uniform discrepancy also gives shrinking-target relative asymptotics when the Sato–Tate measure of the moving target dominates the corresponding discrepancy error. In particular, for $h = \sqrt{X}(\log X)^A$ ($A > 1$), both the one-variable and joint results allow $\varphi(q) = o((\log X)^{A-1})$. Description and Preprint Notes This revised preprint establishes conditional one-variable and joint Sato–Tate localization for non-CM elliptic curves at primes lying simultaneously in a moving interval $(X, X+h]$ and a single arithmetic progression $a \pmod q$, with $q$ allowed to vary with $X$. Under GRH for the relevant finite packets of Dirichlet-twisted symmetric-power and Rankin–Selberg $L$-functions, a two-scale smooth plateau and Dirichlet-character recombination before sharp-edge removal yield finite-depth angular discrepancy bounds. At the progression-normalized Cramér scale $h = G\varphi(q)\sqrt{X}\log X$, the results give expected-order occupancy for fixed positive-measure angular intervals and rectangles when $G$ is sufficiently large. With growing spectral depth and $\eta = (1 + \log G)/G + h/(X\log X)$, the discrepancy bounds are $O(\eta^{1/2})$ in one variable and $O(\eta^{1/3})$ for joint rectangles. The revised manuscript removes a separate auxiliary $q \le X^{1/4}$ restriction from the balanced-scale corollaries by deducing the required modulus conditions from $h \le X/2$, and records consequences for shrinking angular targets. The manuscript unifies and extends the author's earlier single-curve and joint-progression preprints. It is an author preprint, not a peer-reviewed or accepted journal article. A related manuscript is currently under review at the Journal of Number Theory; this public revision contains refinements made after the original journal submission and is not necessarily identical to the version in peer review. The analytic results remain conditional on the stated GRH packets. Creator Lee Byoungwoo Independent Researcher, Daejeon, Republic of Korea Keywords Sato–Tate distribution; non-CM elliptic curves; arithmetic progressions; moving short intervals; Generalized Riemann hypothesis; Rankin–Selberg L-functions; symmetric power L-functions; Dirichlet twists; explicit formula; discrepancy bounds; Cramér scale; shrinking targets