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We consider the class S(λ, β, τ) of convergent for all x ≥ 0 Taylor-Dirichlet type series of the form F(x) = X+∞ n=0 bnexλn+τ(x)βn, bn ≥ 0 (n ≥ 0), where τ: [0,+∞) → (0,+∞) is a continuously differentiable non-decreasing function, λ = (λn) and β = (βn) are such that λn ≥ 0, βn ≥ 0 (n ≥ 0). In the paper we give a partial answer to a question formulated by Salo T.M., Skaskiv O.B., Trusevych O.M. on International conference "Complex Analysis and Related Topics" (Lviv, September 23-28, 2013) ([2]). We prove the following statement: For each increasing function h(x): [0,+∞) → (0,+∞), h′(x) ↗ +∞ (x → +∞), every sequence λ = (λn) such that X+∞ n=0 1 λn+1 - λn < +∞ and for any non-decreasing sequence β = (βn) such that βn+1 - βn ≤ λn+1 - λn (n ≥ 0) there exist a function τ (x) such that τ ′(x) ≥ 1 (x ≥ x0), a function F ∈ S(α, β, τ), a set E and a constant d > 0 such that h-meas E := R E dh(x) = +∞ and (∀x ∈ E): F(x) > (1 + d)μ(x, F), where μ(x, F) = max{|an|exλn+τ(x)βn: n ≥ 0} is the maximal term of the series. At the same time, we also pose some open questions and formulate one conjecture. [ABSTRACT FROM AUTHOR]

Let λ2, λ3, λ4, λ5 be nonzero real numbers, not all negative. Let V be a well-spaced sequence. Assume that λ2/λ3 is irrational and algebraic, and δ > 0. Let E V , N , δ be the number of υ ∈ V with υ ≤ N such that the Diophantine inequality λ 2 p 2 2 + λ 3 p 3 3 + λ 4 p 4 4 + λ 5 p 5 5 - υ < υ - δ has no solution in primes p2, p3, p4, p5. In this paper, we prove that for any ε > 0 , E V , N , δ ≪ N 1 - 19 / 378 + 2 δ + ε , which refines the previous result. [ABSTRACT FROM AUTHOR]

The author has recently introduced the class of CNED sets in Euclidean space, generalizing the classical notion of NED sets, and shown that they are quasiconformally removable. A set E is CNED if the conformal modulus of a curve family is not affected when one restricts to the subfamily intersecting E at countably many points. We prove that several classes of sets that were known to be removable are also CNED , including sets of σ -finite Hausdorff (n - 1) -measure and boundaries of domains with n-integrable quasihyperbolic distance. Thus, this work puts in common framework many known results on the problem of quasiconformal removability and suggests that the CNED condition should also be necessary for removability. We give a new necessary and sufficient criterion for closed sets to be (C)NED. Applying this criterion, we show that countable unions of closed (C)NED sets are (C)NED. Therefore we enlarge significantly the known classes of quasiconformally removable sets. [ABSTRACT FROM AUTHOR]