Abstract
For a primep, we obtain an upper bound on the discrepancy of fractionsr/p, whererruns through all of roots modulopof all monic univariate polynomials of degreedwhose vector of coefficients belongs to ad-dimensional boxℬ. The bound is nontrivial starting with boxesℬof size|ℬ|≥pd/2+ɛfor any fixedɛ<0and sufficiently largep.
Highlights
For an integer m and a polynomial f (X) ∈ Z[X], we consider the set of fractions m, f = r m | f (r) ≡, ≤ r m − (1.1)
We obtain an upper bound on the discrepancy of fractions r/ p, where r runs through all of roots modulo p of all monic univariate polynomials of degree d whose vector of coefficients belongs to a d-dimensional box Ꮾ
The bound is nontrivial starting with boxes Ꮾ of size |Ꮾ| ≥ pd/2+ε for any fixed ε < 0 and sufficiently large p
Summary
For an integer m and a polynomial f (X) ∈ Z[X], we consider the set of fractions. that is, the set of fractions r/m where r runs through all distinct roots of the congruence f (r) ≡ 0(mod m). For an integer m and a polynomial f (X) ∈ Z[X], we consider the set of fractions. Hooley [1] has proved that for any irreducible polynomial f (X) ∈ Z[X], the sequence ᏹ f (X) of all fractions r/m ∈ m, f taken over all nonnegative integers m ≤ X, that is, ᏹ f (X) =. Is asymptotically uniformly distributed in the [0,1] interval when X → ∞, the bound on the discrepancy of the sequence ᏹ f (X) is rather weak. For quadratic polynomials f a stronger bound on the discrepancy has been obtained using a different method by Hooley [2], see [3, 4] for further references to more recent improvements and applications
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