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nt.quantity concept – Does each geometric sequence acquire a diminutive rest modulo a sizable prime?

The require query I’m thinking about is the next.

Fix a diminutive $varepsilonin(0,1)$ and an integer $qge 2$ (you might occupy that $q$ is prime if it helps although I consider it should not signify too mighty). For a sizable prime $P$ and an integer $ainmathbb Z$, outline $G(a,P)={aq^mmod P: m=0,1,2,dots}$ the place the remainders are taken within the meander $(-P/2, P/2)$ (i.e., with the minimal workable absolute worth).

Is it undoubted that for all primes $P$ outdoors of a clique of density at most $varepsilon$ (in any sense of the phrase “density” that’s subadditive), $G(a,P)$ comprises a rest within the meander $(-varepsilon P,varepsilon P)$ for each altenative of $ain mathbb Z$?

However I’ll breathe too thinking about any nontrivial ends in the identical path plane in the event that they fall considerably in need of a whole respond (breathe it affirmative or traverse).

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