Vol 303, No 1 (2018)

In a finite-dimensional Banach space, a closed set with lower semicontinuous metric projection is shown to have a continuous selection of the near-best approximation operator. Such a set is known to be a sun. In the converse question of the stability of best approximation by suns, it is proved that a strict sun in a finite-dimensional Banach space of dimension at most 3 is a P-sun, has a contractible set of nearest points, and admits a continuous ε-selection from the operator of near-best approximation for any ε > 0. A number of approximative and geometric properties of sets with lower semicontinuous metric projection are obtained.

We prove that in the real space l₂(ℤ) of two-sided sequences there is an element such that the sums of its shifts are dense in all real spaces l_p(ℤ), 2 ≤ p < ∞, as well as in the real space c₀(ℤ).

Let \(\mathbb{F}_p\) be the field of residue classes modulo a large prime number p. We prove that if \(\mathcal{G}\) is a subgroup of the multiplicative group \(\mathbb{F}_p^*\) and if \(\mathcal{I} \subset \mathbb{F}_p\) is an arithmetic progression, then \(|\mathcal{G} \cap \mathcal{I}| = (1 + o(1))|\mathcal{G}|\mathcal{I}|/p + R\), where \(|R| < (|\mathcal{I}|^{1/2} + |\mathcal{G}|^{1/2} + |\mathcal{I}|^{1/2}|\mathcal{G}|^{3/8}p^{-1/8})p^{o(1)}\). We use this bound to show that the number of solutions to the congruence xⁿ ≡ λ (mod p), x ∈ ℕ, L < x < L + p/n, is at most p^{1/3−1/390+o(1)} uniformly over positive integers n, λ and L. The proofs are based on results and arguments of Cilleruelo and the author (2014), Murphy, Rudnev, Shkredov and Shteinikov (2017) and Bourgain, Konyagin, Shparlinski and the author (2013).

In 1939 P. Turán started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm 1 on \(\mathbb{I}:= [-1,1]\) (interval) and \(\mathbb{D}:= \{z \in \mathbb{C}: |z| \leq 1\}\) (disk) under the normalization condition that the zeroes of the polynomial in question all lie in \(\mathbb{I} \) or \(\mathbb{D} \), respectively. For the maximum norm he found that with n:= deg p tending to infinity, the precise growth order of the minimal possible derivative norm is √n for \(\mathbb{I} \) and n for \(\mathbb{D} \). J. Erőd continued the work of Turán considering other domains. Finally, about a decade ago the growth of the minimal possible ∞-norm of the derivative was proved to be of order n for all compact convex domains. Although Turán himself gave comments about the above oscillation question in L^q norms, till recently results were known only for \(\mathbb{D} \) and \(\mathbb{I} \). Recently, we have found order n lower estimations for several general classes of compact convex domains, and conjectured that even for arbitrary convex domains the growth order of this quantity should be n. Now we prove that in L^q norm the oscillation order is at least n/log n for all compact convex domains.

We introduce the notion of a weight-almost greedy basis and show that a basis for a real Banach space is w-almost greedy if and only if it is both quasi-greedy and w-democratic. We also introduce the notion of a weight-semi-greedy basis and show that a w-almost greedy basis is w-semi-greedy and that the converse holds if the Banach space has finite cotype.

Closely related notions of the Kolmogorov width and the approximate rank of a matrix are considered. New estimates are established in approximation problems related to the width of the set of characteristic functions of intervals; the multidimensional case (characteristic functions of parallelepipeds) is also considered.

Let \(U(\mathbb{T})\) be the space of all continuous functions on the circle \(\mathbb{T}\) whose Fourier series converges uniformly. Salem’s well-known example shows that a product of two functions in \(U(\mathbb{T})\) does not always belong to \(U(\mathbb{T})\) even if one of the factors belongs to the Wiener algebra \(A(\mathbb{T})\). In this paper we consider pointwise multipliers of the space \(U(\mathbb{T})\), i.e., the functions m such that mf ∈ \(U(\mathbb{T})\) whenever f ∈ \(U(\mathbb{T})\). We present certain sufficient conditions for a function to be a multiplier and also obtain some Salem-type results.

Let G = (G, +) be a compact connected abelian group, and let μ_G denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath, Raikov, and Shields) establishes the bound μ_G(A + B) ≥ min(μ_G(A) + μ_G(B), 1) whenever A and B are compact subsets of G, and A + B:= {a + b: a ∈ A, b ∈ B} denotes the sumset of A and B. Clearly one has equality when μ_G(A) + μ_G(B) ≥ 1. Another way in which equality can be obtained is when A = φ⁻¹(I) and B = φ⁻¹(J) for some continuous surjective homomorphism φ: G → ℝ/ℤ and compact arcs I, J ⊂ ℝ/ℤ. We establish an inverse theorem that asserts, roughly speaking, that when equality in the above bound is almost attained, then A and B are close to one of the above examples. We also give a more “robust” form of this theorem in which the sumset A + B is replaced by the partial sumset A +_εB:= {1A * 1B ≥ ε} for some small ε > 0. In a subsequent paper with Joni Teräväinen, we will apply this latter inverse theorem to establish that certain patterns in multiplicative functions occur with positive density.

A new notion of weak monotonicity of sets is introduced, and it is shown that an approximatively compact and weakly monotone connected (weakly Menger-connected) set in a Banach space admits a continuous additive (multiplicative) ε-selection for any ε < 0. Then a notion of weak monotone connectedness (weak Menger connectedness) of sets with respect to a set of d-defining functionals is introduced. For such sets, continuous (d⁻¹, ε)-selections are constructed on arbitrary compact sets.

We obtain a nontrivial lower bound for the size of the set A/B, where A and B are subsets of the interval [1,Q].