Khintchine’s theorem with rationals coming from neighborhoods in different places

The Duffin–Schaeffer Conjecture answers a question on how well one can approximate irrationals by rational numbers in reduced form (an imposed condition) where the accuracy of the approximation depends on the rational number. It can be viewed as an analogue to Khintchine’s theorem with the added restriction of only allowing rationals in reduced form. Other conditions such as numerator or denominator a prime, a square-free integer, or an element of a particular arithmetic progression, etc. have also been imposed and analogues of Khintchine’s theorem studied. We prove versions of Khintchine’s theorem where the rational numbers are sourced from a ball in some completion of [Formula: see text] (i.e. Euclidean or [Formula: see text]-adic), while the approximations are carried out in a distinct second completion. Finally, by using a mass transference principle for Hausdorff measures, we are able to extend our results to their corresponding analogues with Haar measures replaced by Hausdorff measures, thereby establishing an analogue of Jarník’s theorem.

Tripled Fixed Points and Existence Study to a Tripled Impulsive Fractional Differential System via Measures of Noncompactness

In this paper, a tripled fractional differential system is introduced as three associated impulsive equations. The existence investigation of the solution is based on contraction principle and measures of noncompactness in terms of tripled fixed point and modulus of continuity. Our results are valid for both Kuratowski and Hausdorff measures of noncompactness. As an application, we apply the obtained results to a control problem.

Compact operators on sequence spaces associated with the Copson matrix of order α

In this work, we study characterizations of some matrix classes $(\mathcal{C}^{(\alpha )}(\ell ^{p}),\ell ^{\infty })$ ( C ( α ) ( ℓ p ) , ℓ ∞ ) , $(\mathcal{C}^{(\alpha )}(\ell ^{p}),c)$ ( C ( α ) ( ℓ p ) , c ) , and $(\mathcal{C}^{(\alpha )}(\ell ^{p}),c^{0})$ ( C ( α ) ( ℓ p ) , c 0 ) , where $\mathcal{C}^{(\alpha )}(\ell ^{p})$ C ( α ) ( ℓ p ) is the domain of Copson matrix of order α in the space $\ell ^{p}$ ℓ p ($0< p<1$ 0 < p < 1 ). Further, we apply the Hausdorff measures of noncompactness to characterize compact operators associated with these matrices.

HAUSDORFF MEASURES OF A CLASS OF MORAN SETS

Let [Formula: see text] be the class of Moran sets with integer [Formula: see text] and real [Formula: see text] satisfying [Formula: see text]. It is well known that the Hausdorff dimension of any set in this class is [Formula: see text]. We show that for any [Formula: see text], [Formula: see text] where [Formula: see text] denotes [Formula: see text]-dimensional Hausdorff measure of [Formula: see text]. For any [Formula: see text] with [Formula: see text] there exists a self-similar set [Formula: see text] such that [Formula: see text].