In this paper, we investigate a packet access scheme that is able to support mixed traffics in the presence of high propagation delays. Referring to a Time-Code Division Multiple Access air interface, we propose a Medium Access Control (MAC) protocol based on a random access scheme. A successful attempt grants the use of a slot-code resource. This protocol is named Adaptive Time Code-Packet Reservation Multiple Access (ATC-PRMA), since the access parameters are changed, depending on the traffic load conditions, so as to fulfil Quality of Service requirements. Numerical examples are carried out for the Low Earth Orbit (LEO)-Mobile Satellite System (MSS) scenario, but all these considerations could be applied to High-Altitude Platform Stations (HAPSs) as well. In both cases, high propagation delays prevent an immediate feedback to users. An analytical approach is proposed to study the stability of our MAC scheme. Accordingly, we define a criterion for optimizing system performance. The predicted ATC-PRMA behaviour is supported by simulation results. Finally, we show the performance improvement of ATC-PRMA with respect to a MAC protocol not employing adaptive parameters

We extend Rothberger's theorem (on the equivalence between CH and the existence of Luzin and Sierpiński-sets having power 𝔠) and certain paradoxical constructions due to Erdös. More precisely, by employing a suitable σ-ideal associated to the (α, β)-games introduced by Schmidt, we prove that the Continuum Hypothesis holds if and only if there exist subgroups of (ℝ, +) having power 𝔠 and intersecting every “absolutely losing” (respectively, every meager and null) set in at most countably many points.

Let G be a locally compact Abelian group and μ a Haar measure on G. We prove: (a) If G is connected, then the complement of a union of finitely many translates of subgroups of G with infinite index is μ-thick and everywhere of second category. (b) Under a simple (and fairly general) assumption on G, for every cardinal number m such that ℵ 0 m |G| there is a subgroup of G of index m that is μ-thick and everywhere of second category. These results extend theorems by Muthuvel and Erdős-Marcus, respectively. (b) also implies a recent theorem by Comfort-Raczkowski-Trigos stating that every nondiscrete compact Abelian group G admits 2 |G| -many μ-nonmeasurable dense subgroups.

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