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

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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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