Volume 16 (2021): Issue 1 (June 2021)

In this paper, we classify all solutions with cyclic and semi-cyclic semigroup supports of the functional equations arising from multiplication of quantum integers with fields of coefficients of characteristic zero. This also solves completely the classification problem proposed by Melvyn Nathanson and Yang Wang concerning the solutions, with semigroup supports which are not prime subsemigroups of ℕ, to these functional equations for the case of rational field of coefficients. As a consequence, we obtain some results for other problems raised by Nathanson concerning maximal solutions and extension of supports of solutions to these functional equations in the case where the semigroup supports are not prime subsemigroups of ℕ.

In a family of S_n-fields (n ≤ 5), we show that except for a density zero set, the lower and upper bounds of the Euler-Kronecker constants are −(n − 1) log log d_K+ O(log log log d_K) and loglog d_K + O(log log log d_K), resp., where d_K is the absolute value of the discriminant of a number field K.

We provide an algorithm to approximate a finitely supported discrete measure μ by a measure ν_N corresponding to a set of N points so that the total variation between μ and ν_N has an upper bound. As a consequence if μ is a (finite or infinitely supported) discrete probability measure on [0, 1]^d with a sufficient decay rate on the weights of each point, then μ can be approximated by ν_N with total variation, and hence star-discrepancy, bounded above by (log N)N⁻¹. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure μ, there exist finite sets whose star-discrepancy with respect to μ is at most (log N)d−12N−1 {\left( {\log \,N} \right)^{d - {1 \over 2}}}{N^{ - 1}} . Moreover, we close a gap in the literature for discrepancy in the case d =1 showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.

In the present paper the author uses the function system Γ_ℬsconstructed in Cantor bases to show upper bounds of the extreme and star discrepancy of an arbitrary net in the terms of the trigonometric sum of this net with respect to the functions of this system. The obtained estimations are inequalities of the type of Erdős-Turán-Koksma. These inequalities are very suitable for studying of nets constructed in the same Cantor system.

The higher-dimensional generalization of the weighted q-adic sum-of-digits functions s_q,γ(n), n =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x)= x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function s_q,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h₁s_{q, γ} (n)+h₂s_q,γ (n +1), where h₁ and h₂ are integers such that h₁ + h₂ ≠ 0 and that the akin two-dimensional sequence s_q,γ (n), s_q,γ (n +1) cannot be uniformly distributed modulo one if q ≥ 3. The properties of the two-dimensional sequence s_q,γ (n),s_q,γ (n +1), n =0, 1, 2,..., will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.