References of "Uniform Distribution Theory"
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See detailThe distribution of rational numbers on Cantor's middle thirds set
Trauthwein, Tara UL; Rahm, Alexander; Solomon, Noam et al

in Uniform distribution theory (in press)

We give a heuristic argument predicting that the number N*(T) of rationals p/q on Cantor’s middle thirds set C such that gcd(p, q) = 1 and q ≤ T, has asymptotic growth O(T^{d+ε}), for d = dim C. Our ... [more ▼]

We give a heuristic argument predicting that the number N*(T) of rationals p/q on Cantor’s middle thirds set C such that gcd(p, q) = 1 and q ≤ T, has asymptotic growth O(T^{d+ε}), for d = dim C. Our heuristic is related to similar heuristics and conjectures proposed by Fishman and Simmons. We also describe extensive numerical computations supporting this heuristic. Our heuristic predicts a similar asymptotic if C is replaced with any similar fractal with a description in terms of missing digits in a base expansion. Interest in the growth of N*(T) is motivated by a problem of Mahler on intrinsic Diophantine approximation on C. [less ▲]

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See detailKummer theory for number fields and the reductions of algebraic numbers II
Perucca, Antonella UL; Sgobba, Pietro UL

in Uniform Distribution Theory (2020)

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K*. For almost all primes p of K, we consider the order of the cyclic group (G mod p), and ask whether this number ... [more ▼]

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K*. For almost all primes p of K, we consider the order of the cyclic group (G mod p), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if \ell^e is a prime power and a is a multiple of \ell (and a is a multiple of 4 if \ell=2), then the density of primes p of K such that the order of (G mod p) is congruent to a modulo \ell^e only depends on a through its \ell-adic valuation. [less ▲]

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