Guido Manfredi
email: guido.manfredi@enac.fr

Yannick Jestin

Engie Ineo

An Introduction to Fast Time Simulations for RPAS Collision Avoidance System Evaluation

Keywords: Introduction, Tutorial, Detect And Avoid, Collision Avoidance, Fast Time Simulation

Collision voidance systems are crucial for RPAS integration, yet comparing their performances remain difficult. We believe that using fast time simulations and standard evaluation metrics would facilitate their comparison while providing insight into their benefits. However, fast time simulations are often viewed as hard to set up and limited to large scale demonstrations. We believe even small experiments can take advantage of them with huge benefits. The aim of this work is to ease access to fast time simulations by providing explanations, examples and references to previous works and to free software. We also list commonly used evaluation metrics for collision avoidance system performance ranking. By easing the setup of fast time simulation experiments, we believe future works will be able to provide their results in a more detailed and comparable form.

I. INTRODUCTION

According to the SESAR-JU European Drones Outlook Study [1], the European RPAS market is expected to weight 10 billion euros annually by 2035. The creation of such a large business requires an ongoing regulatory effort. However, before regulation can provide the rules, the research communities need to overcome some technological locks including the Detect And Avoid (DAA) problem. In the context of this work, DAA should be understood as aerial traffic DAA, which involves two functions: Traffic Avoidance (TrA), also called Remain Well Clear (RWC), and Collision Avoidance (CA). However, in controlled airspace, TrA is provided by ATC services so the rest of this paper focuses on the CA function only.

As will be shown in the next section, numerous works have proposed CA algorithms, however their comparison remains difficult because they use different evaluation methods. In this paper, we advocate in favor of using Fast Time Simulations (FTSs) with appropriate validation metrics to assess the effectiveness of a Collision Avoidance System (CAS). In order to ease the use of such method, section III provides an overview of the methodology and the main components of a FTS for CAS. These components need to be integrated in a simulation framework to analyze their interactions, so Section IV introduces three freely available simulation frameworks. Section V presents metrics to post process the FTS output into a performance assessment. Finally, Section VI concludes on the mains reasons to rely on FTS when evaluating a CAS and the main steps to do so.

II. EXISTING EVALUATION METHODS

Simulations and evaluations used in previous works provide useful insight into CAS evaluation methods and what needs to be simulated (flying objects, flight patterns, sensors, etc.). We considered three different sources of information: evaluation from the research communities to show the performances of a given algorithm, evaluation with large scale demonstrator for RPAS integration in civil airspace and evaluations by regulatory bodies to authorize or enforce the use of a CAS for manned aviation. Evaluation methods can be grouped in different categories as proposed by Lacher et al. in [START_REF] Lacher | Unmanned aircraft collision avoidance-technology assessment and evaluation methods[END_REF], where each family of methods addresses specific questions (see Table I). In the following, the term Fast Time Simulation (FTS) is used for simulations where all the agents are computer models implemented by programs with no real-time constraints. In the case of Real Time Simulations (RTS), part of the agents are computer models while some of them are human operators (e.g. pilots, controllers). Finally, in the case of test flights, one or more actual aircraft, equipped with the CAS to be evaluated, are flown along real or simulated traffic.

A. Algorithms Evaluation

Most research algorithm evaluations aim at showing the performances of the collision avoidance or the surveillance part on its own, they rarely consider the whole environment and all its uncertainties. As sensors are scarcely simulated, the first four lines of Table I are rarely considered in these kinds of evaluations. In numerous works, evaluation encounters are designed by hand, usually depicting challenging or worst case scenarios. In [START_REF] Orefice | Aircraft automatic collision avoidance using spiral geometric approach[END_REF], the authors evaluate a CA method based on spiraling motion for avoidance. The evaluation is done on four manually defined scenarios representing challenging geometries, with a point mass model for aircraft. The authors of [START_REF] Sahawneh | 3d path planning for small uas operating in low-altitude airspace[END_REF] perform a simulation with five-degree-of-freedom aircraft models for both ownship and intruders. Encounters are designed by hand and aim at representing typical collision encounters. In [START_REF] Radmanesh | Grey wolf optimization based sense and avoid algorithm for uav path planning in uncertain environment using a bayesian framework[END_REF], the evaluation scenarios are handmade 2-D grids containing moving and/or fixed obstacles, all aircraft being considered as point masses. In order to better assess the effectiveness of a CAS, some works perform FTS: the CAS is evaluated in thousands of simulated encounters. In [START_REF] Lin | Sampling based collision avoidance for uavs[END_REF], the CAS is evaluated by simulating a RPAS flying a pattern in the middle of surrounding traffic obtained through logged ADS-B data. In [START_REF] Weibel | Establishing a risk-based separation standard for unmanned aircraft self separation[END_REF], the authors generate ten million encounters using the MIT uncorrelated encounter model [START_REF] Weinert | Uncorrelated encounter model of the national airspace system, version 2.0[END_REF]. Finally, Allignol et al. simulate more than two hundred thousand encounters based on records of real traffic data and simulated RPAS trajectories [START_REF] Allignol | Integration of uas in terminal control area[END_REF].

In some cases, RTS with a simulated or real flight have been used. The method proposed in [START_REF] Lin | Sampling based collision avoidance for uavs[END_REF] is evaluated with flight tests where the encounter geometries were manually defined. In [START_REF] Lyu | A uav sense and avoid system design method based on software simulation[END_REF], Lyu et al. use the FlightGear JSBsim simulator to integrate a simulated RPAS into real-time acquired ADS-B data.

When it comes to simple CA algorithms evaluation, most works rely on simulations designed by hand, rarely on FTS, RTS or test flights. These methods are more common for larger scale experiments as will be shown in the following.

B. Large Scale Evaluations for RPAS integration

Large scale projects, aimed at evaluating integrated solutions, rely on a large panel of validation tools. Though the following methods are not all completely dedicated to CAS, these evaluation methods are worth mentioning.

The ODREA project [START_REF] Odrea | Operational demonstration of rpas in european airspace (odrea)[END_REF] included FTS, RTS and flight tests. The FTS relied on recorded traffic of one day over the south-west of France. These where replayed and RPAS were simulated as performing missions in the airspace with six different motion patterns, different offsets to the RPAS position and different equipage (TCAS/DAA, coordination) for a total of 874,800 encounters. The RTS involved actual Air Traffic Control Operators (ATCOs) and pseudo-pilots simulating the RPAS and background traffic. Three RPAS (2 slow, 1 fast) where simulated in the middle of traffic in nominal and degraded cases (loss of C2 link). Finally, flight tests involved take-off and landing from/to airports following different routes.

The MIDCAS project [START_REF] Midcas | Mid-air collision avoidance system (midcas)[END_REF] aimed at gaining insight into the efficiency of a DAA solution for TrA and CA. The preliminary simulations consisted in creating a flight area where intruders were spawned at random locations, with fixed speed in order to reach a given density of aircraft. Simulated flights lasted 30s and where run to accumulate 100.000 hours. For the FTS, a Monte Carlo analysis has been carried out on 30.000 encounters generated from the European Encounter Model (primary radar data) and Uncorrelated Encounter Model. The RPAS is simulated as a 6 DOF object. It is worth noting that a pilot behavior model has been used in the simulations. Results were analyzed for statistics over NMAC and Target Level of Safety (TLS). For the RTS, specific short scenarios have been manually designed, with multiple intruders. The ATC simulations where performed by recreating flight plans across three active Airports and CTR in Sweden with flights durations over 40 minutes. The RPAS was commanded by a pseudo pilot and traffic and collision avoidance maneuvers where coordinated by actual ATC operators. Finally, actual flights tests were carried out with more than 90 scenarios (29 virtual and 71 real intruders) to test the traffic and collision avoidance. This work is currently extended in the MIDCAS Standardization Support Phase (MIDCAS SSP) with more FTS and RTS to come.

In a long series of experiments aimed at evaluating different CAS methods, the NASA performed numerous FTS, RTS, pilot and controller evaluations as well as test flights. Preliminary works allowed defining needed notions like sensors requirements [START_REF] Park | Investigating detect-and-avoid surveillance performance for unmanned aircraft systems[END_REF] and a well-clear definition [START_REF] Comstock | Unmanned aircraft systems human-in-the-loop controller and pilot acceptability study: Collision avoidance, self-separation, and alerting times (cassat)[END_REF]. Based on these studies, RPAS integration experiments [START_REF] Murphy | Flight test overview for uas integration in the nas project[END_REF], tested three different DAA algorithms, CPDS (General Atomics and TU Delft), JADEM (NASA Ames) and Stratway+ (NASA Langley) during flight tests. These flight tests mixed control tower and background traffic simulated with the NASA LVC environment and two live aircraft for the ownship and the intruder. Through 11 flights, 108 encounters were created.

These examples show that various simulation types are usually combined for large scale evaluations, a common combination being: FTS, RTS and flight tests. As noted in the MIDCAS documentation, FTSs allow statistical validation, RTSs provide functionalities validation and flight tests give results for actual situations.

C. Evaluation for Regulation

CASs for manned aviation have been around for decades. Before being authorized or enforced, these systems have been thoroughly tested. In the last fifteen years, three important systems have been evaluated: ACAS II, STCA, and ACAS X (in the US only).

During the ACASA project [START_REF] Arino | Studies on the safety of acas ii in europe[END_REF], multiple experiments have been conducted to evaluate the effect of ACAS II on safety depending on various conditions. These evaluations have led to the development and use of the European Encounter model, and focused on the number of encounters as well as the risk ratio.

To establish Short Term Conflict Alert (STCA) performance requirements, a series of FTS have been carried out in [START_REF] Raynaud | Applicability and usefulness of the encounter model-based methodology for safety net assessment[END_REF]. Encounters were generated using the ATM encounter model (440,000 encounters where separation minima are preserved) and the safety encounter model (200,000 encounters with risk of Mid Air Collision).

The recent ACAS X preliminary evaluations were done through FTS followed by test flights. Radar surveillance from 100,000 real encounters that resulted in TCAS RA were used to compare to TCAS performances. To obtain more data, hundreds of thousands of encounters were generated using the correlated and uncorrelated MIT encounter models. Plus, specific models for specific situations (500ft vertical separation, 1000ft vertical separation, closely spaced parallel runways) were developed. Finally, encounters from real flights recorded before the use of TCAS served for stress cases. After tuning the algorithm with the preliminary tests, successive flight campaigns have been carried out to evaluate the system in real flight.

On top of these CAS evaluations, concepts of "well-clear" have been recently evaluated [START_REF] Cook | Defining Well Clear for Unmanned Aircraft Systems[END_REF] and though this is not specifically CAS evaluation, the methodology is similar and so interesting to us. The idea is to define with objective metrics the subjective notion of "well-clear". Three candidate definition from NASA, MIT and AFRL were evaluated and their results compared. The evaluation was performed in three stages. First a Monte-Carlo simulation, then a human-in-theloop simulation and finally a stress case analysis.

Works evaluating new CA algorithms most often rely on hand made worst cases scenarios to show the robustness of their methods. For RPAS large scale evaluations, different simulation methods, from handmade simulations to flight tests, are used to answers different questions about the CAS (see Table I). Finally, for regulation validation, it appears like FTS is the preferred evaluation method as it allows computing statistically significant results, the main drawback being that it requires an encounter model representative of the airspace being assessed. As stated by DeGarmo (MITRE) [START_REF] Degarmo | Issues concerning integration of unmanned aerial vehicles in civil airspace[END_REF] : "How exactly UAV [RPA] operation will affect airspace capacity and traffic flows can only be answered through simulations of UAVs [RPAs] within specified airspace. Part of this simulation must account for how the vehicles will operate (speeds, altitudes, endurance), where they will operate (mission), and in what numbers".

The next section presents the main components of a CAS and the different environmental agents in the FTS that feed them information.

III. FTS FOR CAS

Before going on, we present a general architecture for a CAS and detail its sub-function. Then, we present the modules needed in the FTS to feed information to these sub-functions and simulate all the environment.

A. Collision Avoidance System

In the following the aircraft equipped with the CAS is called the ownship, surrounding aircraft currently monitored are intruders and aircraft requiring avoidance are threats. The goal of a CAS is to prevent Near Mid-Air Collisions (NMACs), i.e. threats entering the ownship collision volume. In order to have time for the avoidance process, a collision avoidance threshold is defined around the collision volume.

To perform this avoidance, a CAS is composed of four different processes which allow processing surveillance data, estimating the current state and next state, detecting possible threats and finding a maneuver to avoid them. These steps are illustrated in Figure 1 and detailed in the following.

1) Detection & Tracking: The CAS surveillance system monitors aircraft within its sensors range. The detection and tracking functions collect sensor measurements to creates tracks. It also has the responsibility to merge the input from the sensors and verify the data provided by sensors deemed unsecure. Depending on the process and sensors involved, a confidence level is associated with each track.

2) State Estimation: From different data provided by the surveillance, the state estimation process computes a set of variables which describes the current situation w.r.t an intruder or a group of intruders. The number of variables should be kept low to reduce computational complexity but descriptive enough to provide a proper view of the current state. For example, in TCAS the state is describe by τ , an estimation of the time before Closest Point of Approach (CPA), and h, the relative altitude between ownship and a threat. A common method to reduce the number of state variables is to use aggregate variables which mix various information into a single variable, e.g. in TCAS τ is an aggregation of the range and closure rate variables. To handle noise in the surveillance data, modern methods consider the state variables as probability distributions.

3) Dynamic Model: The dynamic model allows estimating the future state of the world, at the next time step. It can include ownship/intruder dynamics and pilot response models. Estimation of an intruder's dynamic model can be computed with straight projections, worst case projections, probabilis-projections or with a flight plan sharing approach. The dynamic model computation often depends on the previously estimated state to ensure a memory effect, where past states influence the estimation of future states.

4) Threat Metric: The threat metric is the formal definitions of the CA threshold. It can depend on the ownship-intruder distance, closure rate, time to CPA and/or time to collision volume violation. It can also consider only horizontal metrics or both horizontal and vertical metrics. Often, like in TCAS, the threat metric is a combination of both time and space measures with multiple conditions to be met before declaring an intruder has become a threat.

5) Threat Detection: For each monitored aircraft, the threat metric allows the threat detection process to estimates if there is a collision risk. Based on the current and projected states, and on a set of rules defining the collision avoidance threshold, it determines if an intruder has become a threat.

6) Threat Resolution: When one, or various, intruders are labelled as threats, the threat resolution module computes a maneuver to avoid them. This maneuver can influence three parameters: ownship speed, horizontal direction and vertical direction. The resulting maneuver can be a simple command to follow at a given time or a full trajectory leading out of danger.

The CAS output is communicated to the remote pilot and is executed by the RPA or by the remote pilot (depending on the remote pilot being in-the-loop or on-the-loop). After this quick overview of CAS architectures and workings, let's focus on the evaluation of such algorithms.

B. Fast Time Simulations

Fast Time Simulations (FTS) allow performing a high number of tests with limited logistics, in limited time and with no safety concerns. The different elements of the world need to be modeled, mainly the aircraft traffic and RPASs as well as the sensors used by each one of them. Encounters that will activate the CASs need to be generated. As will be discussed later, there are various means to do that with different objectives in mind.

1) Aircraft Models: There are three popular representations for aircraft. The simplest one is the point mass model, where an aircraft is considered as a point upon which forces are applied. This representation completely hides the flight mechanics. A more complex representation involves considering the aircraft as a 4 degree of freedom object [START_REF] Cook | Defining Well Clear for Unmanned Aircraft Systems[END_REF]. Then an aerodynamic model of the aircraft is needed to represent its behavior depending on these DOFs. Finally, the most complex model uses 6 DOFs, the additional dimensions increase the realism and complexity of the simulation. For the two previous representations, manned aircraft models are available in the Base of Aircraft Data (BADA). Some RPAS models have been created and integrated as well in the BADA database [START_REF] Sanchez | Creation of rpas aircraft performance models for bada[END_REF]. The precision of the aircraft models need to be different depending on the considered constraints. For example, if wind needs to be accounted for, then the point mass model provides a rough approximation. It is important to note that all the aircraft involved in the evaluation do not need precise modelling. A good compromise can be reached by representing the traffic with the point mass model and to have the RPAS modeled with 4 to 6 DOFs.

The model should include as a minimum the following information: maximum climb rate, vertical acceleration, maximum turn rate, and maximum turn acceleration.

2) Pilot Models: The reaction time and decisions of the remote or intruder aircraft pilot can be modeled to increase the simulation's realism. It provides the advisory response rate, and advisory response delay. The simplest model is the procedural pilot, it considers that the pilot follows all the required procedures for his flight without accounting for the surrounding aircraft, except if its onboard system (e.g. TCAS) gives an alert. In this model, no reaction to a maneuvering aircraft will be done. Variations of this model consider the reaction time of a pilot. It can be instantaneous, though it has been shown that this creates discrepancies with real data [START_REF] Theunissen | Well clear recovery for detect and avoid[END_REF]. It can take a fixed amount of time, e.g. ACAS considers that pilots answer resolutions in 5s [START_REF] Eurocontrol | Airborne Collision Avoidance Systems (incorporating TCAS II version 7.0 & 7.1 and introduction to ACAS X)[END_REF]. Or it can use a probabilistic reaction time model, as is done in ACAS X [START_REF] Kochenderfer | Next generation airborne collision avoidance system[END_REF]. Advanced pilot models have been developed, where the pilot's reaction changes depending on the maneuver of surrounding aircraft. This has been modeled using game theoretic models in [START_REF] Musavi | Unmanned aircraft systems airspace integration: A game theoretical framework for concept evaluations[END_REF].

3) Sensor Models: Like for the aircraft models, the sensors can be represented with different levels of detail. The simplest level is to consider the suite of sensors as a whole. For example, in [START_REF] Park | Investigating detect-and-avoid surveillance performance for unmanned aircraft systems[END_REF] the suite of sensors is represented by a single sensor with given range, Field of Regard (FOR) and update rate.

A second approach is to model each sensor independently. This requires the existence of a surveillance module able to perform data fusion to combine the information brought by each sensor. Care need to be taken for dependent sensors to model their source of information. Indeed, the GNSS data used by a plane to estimate its own position need to have the same error as the one it is broadcasting through its ADS-B. Modeling sensors independently has the benefit of differentiating cooperative sensors, i.e. sensors taking advantage of intruder aircraft avionics, from non-cooperative sensors, i.e. sensors relying only on onboard processing for aircraft detection. In the following, we provide for common RPAS sensors, simple models for the GNSS and INS as well as models developed in [START_REF] Lyu | A uav sense and avoid system design method based on software simulation[END_REF] for the Air-To-Air-Radar (ATAR) and Electro Optics/Infra Red (EO/IR).

All aircraft are equipped with Global Navigation Satellite System (GNSS) receivers to acquire their localization, latitude, longitude, altitude, in a global frame. Modeling of a GNSS system can reach high levels of complexity. However, for aircraft positioning such precision is neither required nor desirable as it would be far too complex to simulate. Instead, a simple model consists in considering the three dimensions of the localization as independent and adding Gaussian noise to each of them. Though, this approach does not consider the time correlation between successive measurements, so a more realistic representation can be achieved using a first order Gauss-Markov process, as described in [START_REF] Mohleji | Modeling ads-b position and velocity errors for airborne merging and spacing in interval management application[END_REF]. According to this work,

y(n) = ay(n -1) u(n),
where n is the time (discrete), y(n) is the GNSS measurement at time n, a is a coefficient belonging to ]0; 1[ and u(n) is a random variable that follows a Gaussian distribution. While remaining simple, this method provides a more realistic GNSS representation.

In order to estimate their attitude, aircraft rely on Inertial Navigation System (INS) measurements combined with estimation filters to track their motion. This provides roll, pitch, yaw, the corresponding angular speeds as well as ground speed and vertical speed. Integrating this data allows computing the aircraft localization. Again, the simulation of an INS is the topic of a fair amount of research. Some of the most advanced models take into account the equation of motion of the INS parts to model it, but again this is too complex for the task at hand. In the following, we propose a model combining white noise, a constant bias and first order Markov noise. The resulting noise is computed as follows,

η = σ(1 + 0.5randn(1 + 1 -β 1 + β )/ √ 3)
with σ the quadratic error at 95%, randn a random number from a Gaussian distribution and β a Markov noise constant defined as β = exp -δ t τ where δ is the sampling period and τ a constant specific to the INS. The previous noise is applied independently on the eight parameters provided by the INS.

The Air-To-Air-Radar (ATAR) is a main sensor for medium and large RPAS as it is the only non-cooperative sensor working in IMC while having large FOR and range. According to [START_REF] Lyu | A uav sense and avoid system design method based on software simulation[END_REF], an ATAR model can be simplified by neglecting the signal processing part. Their model consists in projecting the intruder global position X i into a spherical coordinates system S centered on the ownship global position X o and adding zero mean Gaussian noise independently on the three dimensions. With X io = X i -X o = (x io , y io , z io ) and S io = (r io , α io , β io ) the expression of X io in S:

r io = x 2 io + y 2 io + z 2 io + η r , η r ∼ N (0, σ r ) θ io = arctan y io x io + η θ , η θ ∼ N (0, σ θ ) φ io = arccos z io r io + η φ , η φ ∼ N (0, σ φ )
where (η r , η θ , η φ ) represents the Gaussian noise. So the output of the ATAR is the (r io , α io , β io ) coordinates with associated rates and variances.

Because it is a cheap sensor that allows a great deal of different processes, Electro Optics and Infra Red (EO/IR) are generally present in the RPAS sensor suite. In order to operate by day and by night, visual spectrum cameras are coupled with Infra Red cameras. Simulating such sensors in detail would require a photorealistic simulator able to generate realistic synthetic images of the simulation environment. This can be done thanks to COTS software, like in [START_REF] Lyu | A uav sense and avoid system design method based on software simulation[END_REF] where the authors used the Vega Prime software [START_REF] Vega Prime | [END_REF] to simulate both EO and IR sensors. Proper simulation of the IR sensor implies some more processing than EO and a possible method is described in [START_REF] Yu | Application of dynamic ir scene generation technique in airborne electro-optical detecting simulation systems[END_REF]. Once images are simulated, image processing techniques, like the ones described in [START_REF] Mcfadyen | A survey of autonomous vision-based see and avoid for unmanned aircraft systems[END_REF], can be used to retrieve the intruder bearing and relative altitude. However, retrieving precise range would require a complicated setup which is rarely considered. Simulated sensors introduces noise in the traffic position measurements which need to be handled with a probabilistic representation of the environment in the CASs to be evaluated. In order to keep the noise in realistic boundaries, it is crucial to consider the minimum requirement on some equipment. For example, simulated ADS-B data should conforme to the minimum navigation precision described in existing standards (NIC, NACp, NACv, SIL levels). Similarly, active surveillance (ACAS) should conforme to ACAS's MOPS (DO-185B). With models for aircraft, RPAS and their sensors, it is time to proceed to the encounter models, i.e. how to simulate conflicts between a RPAS and traffic in a realistic fashion.

4) Encounter Model: The encounter model defines the parameters of the conflict between a RPAS and one or various aircraft. The objective is to create realistic encounters while covering as many cases as possible. Depending on the goal of the evaluation, different types of models can be used: safety critical (for safety analysis), operational suitability (for acceptability), stressing (for late stage evaluations), specific (to be sure the system can handle some precise scenarios). Multiple encounter models have been built for manned aviation by analyzing radar tracks (see Table II). However, this is not possible in the RPAS case due to the lack of data concerning how RPAS fly in the airspace. Nevertheless, we will present three different approaches to deal with the encounter modeling problem.

First, the parameterized methods rely on hand designed encounters with a parameterization done by experts. This type of approach is useful to define stress cases which are unlikely to happen during an actual flight but need to be handled nonetheless. The AFRL has defined such stress case encounters when evaluating "well clear" definitions [START_REF] Cook | Defining Well Clear for Unmanned Aircraft Systems[END_REF]. Because of the limited number of encounters that can be designed manually, this approach offers the possibility to examine in fine detail each encounter and its development. This is especially useful to get an insight of the CAS behavior in abnormal and emergency situations. As an example, the authors of [START_REF] Zou | On the validation of a uav collision avoidance system developed by model-based optimization:: Challenges and a tentative partial solution[END_REF] propose to generate encounters with evolutionary computation methods to create difficult cases.

The probabilistic approach consists in using real traffic data to estimate a probability distribution over the encounter parameters. Once this distribution is estimated, generating an encounter is as simple as sampling the distribution. This approach is often called "Monte Carlo" simulation, though we prefer to avoid this name as the Monte Carlo method can be used to compute different type of data in simulations. II: Existing encounter models for manned civil aviation, built from primary radar tracks. This solution may be the most popular because of its realism and its capacity to generate as many encounters as needed. Some popular probabilistic encounter models usable for RPAS include [START_REF] Kochenderfer | Correlated encounter model for cooperative aircraft in the national airspace system version 1.0[END_REF], [START_REF] Weinert | Uncorrelated encounter model of the national airspace system, version 2.0[END_REF]. It can even be integrated in the DAA process, indeed the authors of [START_REF] Kuchar | A review of conflict detection and resolution modeling methods[END_REF] proposed to use the probabilistic model obtained with such method directly in a DAA algorithm to estimate the risk associated with a maneuver in real time.

The last encounter model, and maybe the most realistic, is a mix between real traffic data and simulated RPAS data. The idea is to define RPAS missions over a given territory and simulate the RPAS flight during these missions. Then, recorded traffic is overlaid on the given territory. By replaying the recorded traffic and making the RPAS follow their missions, encounters are generated. This method has been used in [START_REF] Park | Investigating detect-and-avoid surveillance performance for unmanned aircraft systems[END_REF], with socio-economical studies defining eight mission types over the whole U.S. territory. It has also been used in the ODREA project [START_REF] Odrea | Operational demonstration of rpas in european airspace (odrea)[END_REF] with missions defined in the vicinity of an airport to observe encounters between low-level RPAS and arriving/departing aircraft.

Depending on the type of traffic being considered, two types of encounters are possible: correlated and uncorrelated. In correlated encounters, the aircraft involved are cooperative and at least one of them is in contact with ATC. This implies that their motions are linked to some extent through the intervention of ATC. In uncorrelated encounters, at least one of the aircraft is non-cooperative, i.e. not providing its position. The type of encounter impacts the aircraft trajectories as well as the RPAS sensor suite. Indeed, correlated encounters resemble IFR/IFR flights encounters while uncorrelated encounters resemble IFR/VFR flights encounters. For detailed sensors simulation, in correlated encounters the RPAS needs only cooperative sensors while for uncorrelated encounters the RPAS requires cooperative and non-cooperative sensors.

IV. EXISTING SIMULATION FRAMEWORKS

The previous Section listed the required elements to perform an FTS for CAS performance evaluation. However, for these elements to live, synchronize and communicate together, a simulation framework is needed. In the following, we introduce three freely available simulation frameworks to perform FTS: InCAS, from EUROCONTROL; ACES, from NASA Ames; EASY, from DSNA.

A. InCAS (to become CAFE)

Developed by EUROCONTROL, the InCAS framework [34] allows replaying traffic and analyzing it for CAS performance estimation. It allows analyzing close encounters individually for incident investigations. InCAS processes radar data to provide clean trajectories with a one-second interval. A TCAS module allows simulating collision avoidance logic in the manned airplanes.

A new framework is currently under development, the Collision Avoidance Fast-time Evaluation (CAFE), for airborne and ground-based applications. It will include a close encounter model of the European airspace and rely on rare events simulation methods. Its develoment is being supported by both EUROCONTROL and EUROCAE WG-75, along with SESAR-JU projects.

B. ACES

The Airspace Concept Evaluation System (ACES) is a gateto-gate simulation architecture developed by NASA Ames [START_REF] George | Build 8 of the airspace concept evaluation system[END_REF]. It can replay recorded traffic at a National Airspace System (NAS) or local scale and simulate airspace, aircraft and wind behavior. Its agent based modeling approach provide an independent description for each actor of the airspace. Communications between all agents are available and can be analyzed. A large library of JAVA modules can be plugged on top of the core ones, and custom modules can be added. The execution is sped up by its capability to distribute multiple simulations on a set of machines.

C. EASY

EASY is the simulation framework developed and used by the DSNA, the French Air Navigation Service Provider (ANSP) [START_REF][END_REF]. It can replay traffic at an international scale with trajectories computed with an adjustable timescale which can go under the second (e.g. for high update rate sensors). Its modular architecture allows plugging-in new modules regardless of the language used to develop them and without needing to access their code (black box modules). A library of existing modules (e.g. pilot behavior, TCAS logic) allows quickly building a full simulation environment. Finally, "as fast as possible" logics and parallelization on multiple machines provide quick computation for large FTS. This list is not exhaustive, more simulation frameworks can be found in the literature and in the commerce (e.g. CASSAT, SIMMOD, AirTOp, Flames, etc.).

V. VERIFICATION AND VALIDATION

Based on the FTS outputs, through different metrics, verification and validation give objective marks to the CAS being evaluated. According to the E-OCVM3 validation methodology [START_REF]E-ocvm version 3.0 volume i: European operational concept validation methodology[END_REF], Verification deals with answering the question "Are we building the system right?". On the other hand, Validation deals with "Are we building the right system?".

A. Verification

A comprehensive way to a system is to use formal verification methods. By modelling a system and checking that this model always respects given rules it is possible to ensure that the system will always behave as expected. Though highly effective, these methods require a lot of work to set up. Moreover, as noted in [START_REF] Hagen | Stratway: A modular approach to strategic conflict resolution[END_REF], they ask for the system to follow particular designs. Example of such approaches are available in [START_REF] Jeannin | A formally verified hybrid system for the next-generation airborne collision avoidance system[END_REF], [START_REF] Muñoz | Daidalus: detect and avoid alerting logic for unmanned systems[END_REF], [START_REF] Muñoz | Unmanned aircraft systems in the national airspace system: a formal methods perspective[END_REF], [START_REF] Von Essen | Probabilistic verification and synthesis of the next generation airborne collision avoidance system[END_REF].

A more empirical, nevertheless effective, way to verify a CAS is to simulate multiple encounters and verify if the system behaves as expected. This kind of approach requires a large number of encounters in order to explore as many as possible of the situations that could arise. In [START_REF] Zou | On the validation of a uav collision avoidance system developed by model-based optimization:: Challenges and a tentative partial solution[END_REF], the authors propose to use evolutionary computation to create encounters from a given encounter model and search for difficult cases to explore from the simplest to the hardest cases.

B. Validation

Various actors are impacted by the behavior of a CAS: the public, RPAS operators, remote pilots, pilots, ATM operators, each one with different expectations. This is reflected through different metrics used to validate a CAS. According to Holland et al. [START_REF] Holland | Optimizing the next generation collision avoidance system for safe, suitable, and acceptable operational performance[END_REF], these metrics can be divided into three categories: safety, operational and acceptability metrics.

Safety metrics are directly related to the capacity of the CAS to prevent NMACs. Operational metrics measure the disruption of the avoidance maneuver on the different airspace actors, including ATC and the other aircraft. Acceptability metrics relate to the remote pilot expectations. These metrics reflect how much a remote pilot can trust the system to act as the pilot would have.

The most common safety metric is the risk ratio; it compares the probability of NMAC with and without the CAS. It does not reflect the intrinsic safety of the CAS but rather the safety gains compared to no/another system. In facts, the risk ratio is composed of two quantities: the risk reduction from the CAS and the risk induced by the CAS. Indeed, the CAS may alert when not necessary and the resulting maneuver may increase the risk compared to no CAS. In the case of RPAS, it has been noted in [START_REF] Londner | Collision avoidance system effectiveness on low performance unmanned aircraft[END_REF] that, due to their lower climbing rate, encounters range rate wherer lower and so was the risk ratio. In order to factor out this element, a modified risk ratio is proposed which consider the RPAS as having climb rates similar to the rest of the traffic when computing the denominator of the risk ratio. Table III lists some common validation metrics. More details can be found in [START_REF] Lee | Evaluating alerting and guidance performance of a uas detect-and-avoid system[END_REF], [START_REF] Holland | Optimizing the next generation collision avoidance system for safe, suitable, and acceptable operational performance[END_REF], [START_REF] Cook | Defining Well Clear for Unmanned Aircraft Systems[END_REF], [START_REF] Espindle | Safety analysis of upgrading to tcas version 7.1 using the 2008 us correlated encounter model[END_REF], [START_REF] Midcas | Mid-air collision avoidance system (midcas)[END_REF], [START_REF] Odrea | Operational demonstration of rpas in european airspace (odrea)[END_REF].

VI. CONCLUSIONS

Often regarded as a cumbersome evaluation method, FTS can be set up quickly thanks to existing tools and results. With the proper validation metrics, they provide rich insight in the pros and cons of a CAS. This paper gathered multiple results and references to allow the reader to quickly start using FTS and adapt it to its needs. Future work aims at creating a benchmark widely available with a plug-and-play feature to allow CAS developers to compare their system on an identical setup.

Fig. 1 :

 1 Fig. 1: The five main steps of a collision avoidance system.

TABLE I :

 I Evaluation Methods, and the Questions That Can Be Addressed (Table from[START_REF] Lacher | Unmanned aircraft collision avoidance-technology assessment and evaluation methods[END_REF]). In this Table, the Fast Time Simulation methods are called Monte Carlo Simulations.

ACKNOWLEDGMENT

This work was supported by the ENGIE Ineo -Groupe ADP -Safran RPAS Chair. The authors would like to thank Catherine Ronflé-Nadaud, Julien Farjon, Eric Blond and Christian Aveneau for sharing their extensive knowledge on the topic.

III: Example of validation metrics commonly used for CAS safety assessments.

Safety Mitigated risk ratio N M AC CAS N M AC noCAS

Denote the safety gain when using the proposed CAS.

Modified risk ratio

Similar to mitigated risk ratio, but factors out the fact that RPAS climb slowly. Relative risk ratio

Denote the safety gain when using CAS1 instead of CAS2.

Unresolved risk ratio

This number reflects the proportion of encounters unresolved by the CAS. Risk ratio = unresolved risk ratio + induced risk ratio. Induced risk ratio

This number reflects the proportion of encounters created by maneuvers provided by the CAS. Risk ratio = unresolved risk ratio + induced risk ratio.

Positive predictive value

T P T P +F P

The probability of declaring an alert when necessary relative to the total number of alerts. False discovery rate

F P T P +F P

The probability of declaring an alert when unnecessary relative to the total number of alerts. True Positive (TP) rate

The probability of declaring an alert when a collision risk exists.

False Positive (FP) rate

The probability of declaring an alert when no collision risk exists. Provides an insight on how close the avoidance takes place and if the resulting maneuver has been successful.

Missed Alert rate


Emma D'aniello
email: emma.daniello@unicampania.it.

Laurent Moonens
email: laurent.moonens@math.u-psud.fr.

Joseph M Rosenblatt

Emma D ' Aniello

DIFFERENTIATING ORLICZ SPACES WITH RARE BASES OF RECTANGLES

Keywords: Mathematics Subject Classification, Primary 42B25; Secondary 26B05, 1

In the current paper, we study how the speed of convergence of a sequence of angles decreasing to zero influences the possibility of constructing a rare differentiation basis of rectangles in the plane, one side of which makes with the horizontal axis an angle belonging to the given sequence, that differentiates precisely a fixed Orlicz space.

In the sequel we always call interval in R 2 a set of the form Q = [a, b] × [c, d] where a < b and c < d are real numbers; a rectangle, on the other hand, will be any set obtained from an interval by some isometry of the plane.

There is a long history of research around the maximal operator M θ associated to a set θ ⊆ [0, 2π) of angles, defined as:

M θ f (x) := sup R 1 |R| R |f |,
where the upper bound extends to all rectangles R in R 2 containing x, one side of which make an angle θ ∈ θ with the horizontal axis. It has been shown in 1977 by A. Cordoba and R. Fefferman [START_REF] Córdoba | On differentiation of integrals[END_REF] that whenever θ = {θ j } is the image of a lacunary sequence (i.e. satisfying θ j+1 λθ j for some 0 < λ < 1), then M θ is bounded on L 2 (R 2 ) -and hence also on L p (R 2 ) for all 2 p < ∞. This was extended to any 1 < p < ∞ a year later by A. Nagel, E. M. Stein and S. Wainger in [START_REF] Nagel | Differentiation in lacunary directions[END_REF]. When θ = {0} ∪ {1/j : j ∈ N * }, M. de Guzman proved in 1981 (see [START_REF] De Guzmán | Real Variable Methods in Fourier Analysis[END_REF]) that M θ is always unbounded on L p (R 2 ) for any p > 1. Later, many authors considered the influence that the "size" of the set θ has on its yielding the (un)boundedness of M θ on some L p spaces; a case of particular interest was that of Cantor sets, dealt with by e.g. N. Katz in 1996 [START_REF] Katz | A counterexample for maximal operators over a Cantor set of directions[END_REF] and K. Hare in 2000 [START_REF] Hare | Maximal operators and Cantor sets[END_REF]. Lately, in 2009, M. Bateman gave in [START_REF] Bateman | Kakeya sets and directional maximal operators in the plane[END_REF] a beautiful characterization of sets of angles θ yielding the boundedness of M θ in L p (R 2 ) for all 1 < p < ∞, showing it is equivalent to the possibility of covering θ by a finite collection of N -lacunary sets (see [START_REF] Bateman | Kakeya sets and directional maximal operators in the plane[END_REF] for a definition). Even more recently in 2013, P. Hagelstein studied in [START_REF] Hagelstein | Maximal operators associated to sets of directions of Hausdorff and Minkowski dimension zero[END_REF] the relation between the Minkowski dimension of θ being zero, and the boundedness in L p (R 2 ) of the associated maximal operator; namely he shows that it is a necessary, but not sufficient condition for its boundedness.

A first observation we make in the current paper is to observe that the boundedness of M θ either occurs in L p (R 2 ) for all 1 < p ∞, or fails to occur in any of those spaces. The fact that one can take p = ∞ in the latter statement being, as it seems to us, a new observation following from Bateman's work [START_REF] Bateman | Kakeya sets and directional maximal operators in the plane[END_REF]. We expose our short argument in section 2.

On another hand, the boundedness on L p of the maximal operator M θ is also related to differentiation properties of the associated differentiation basis; namely M θ is bounded on L p (R 2 ) for all 1 < p < ∞ if and only if Lebesgue's differentiation theorem holds in all L p (R 2 ) spaces (1 < p < ∞) for the associated basis, i.e. if and only the following equality holds for all f ∈ L p (R 2 ) (1 < p < ∞) and a.e. x ∈ R 2 :

f (x) = lim R x diam R→0 1 |R| R |f |,
where the limit is taken on rectangles R containing x, one side of which makes an angle θ ∈ θ with the horizontal axis; we then say that the associated basis differentiates L p (R 2 ) for all 1 < p < ∞ (see e.g. [START_REF] De Guzmán | Differentiation of Integrals in R n[END_REF]Chapter III] where it is studied how the differentiation of function spaces by a basis, and the behavior of the associated maximal operator are related). It is also the result of an observation by J.-O. Strömberg in [START_REF] Strömberg | Weak estimates on maximal functions with rectangles in certain directions[END_REF] that for θ = {2 -k : k ∈ N}, the above differentiation theorem fails in any Orlicz space "larger" than L log 2 L(R 2 ), i.e. in any

L Φ (R 2 )
where Φ is an Orlicz function (see below for the precise definition of those term and space) satisfying Φ(t) = o(t log 2 + t) at ∞. In the present paper, we also study for various sequences (θ j ) decreasing to 0, the differentiation properties of some differentiation bases of rectangles in R 2 whose elements have countably many shapes (i.e. for which the ratio between the length of the horizontal and vertical sides belong to a countable set), knowing moreover that one of their sizes make an angle θ j with the horizontal axis for some j, in terms of the speed of convergence of the sequence (θ j ).

More precisely, we prove three theorems of the following structure, for three classes of sequences θ = (θ j ) decreasing to 0, where Φ is some Orlicz function and L Φ (R 2 ) is the associated Orlicz space (see below for the precise definitions of the terms used here as well as in the following statement).

Theorem 1. There exists a countable family Q of intervals of the form [0, L] × [0, l] having the following properties: Namely, we consider the three following cases in the previous statement:

(i) the differentiation basis B of all intervals in R 2 whose shape is that of some Q ∈ Q, differentiates L 1 (R 2 ); (
(1) θ = (θ j ) satisfying 0 < lim j (θ j+1 /θ j ) lim j (θ j+1 /θ j ) < 1 -in which case the statement above holds for Φ(t) = t(1

+ log + t) and L Φ (R 2 ) = L log L(R 2 ) (see Theorem 10); (2) θ = (θ j ) satisfying 0 < lim j (θ j+1 /θ d j ) lim j (θ j+1 /θ d j ) < 1 for some integer d > 1 - in which case the statement above holds for Φ(t) = t(1 + log + log + t) and L Φ (R 2 ) = L log log L(R 2 ) (see Theorem 12);
(3) θ = (θ j ) defined by θ j = arctan(a j d ) for some 0 < a < 1 and 0 < d < 1 -in which case the statement above holds for Φ(t) = t(1 + log Theorem 13).

1/d + t) and L Φ (R 2 ) = L log 1/d L(R 2 ) (see
Those results rely on some geometrical preliminaries (detailed in section 3) and on a nice previous work by A. Stokolos [START_REF] Stokolos | On the differentiation of integrals of functions from Orlicz classes[END_REF] concerning differentiation bases of rectangles in relation to the Orlicz spaces they differentiate. More specifically, in [START_REF] Stokolos | On the differentiation of integrals of functions from Orlicz classes[END_REF], A. Stokolos constructs, for Orlicz spaces ranging (roughly speaking) between L log L(R 2 ) and L log 2 L(R 2 ), differentiation bases of rectangles B satisfying the following properties:

(1) rectangles in B have one side forming an angle 2 -k with the horizontal axis for some k ∈ N; (2) B differentiates precisely L Φ (R 2 ). Our point, in comparison to Stokolos' result, is here to consider more general sequences (θ j ) decreasing to 0, and to see how their convergence speed influences the Orlicz space that one can differentiate using rectangles obtained from intervals enjoying given shapes by rotations of some angle θ j .

The structure of the paper is as follows: after studying some geometrical preliminaries in the spirit of [START_REF] Moonens | Differentiating along rectangles, in lacunary directions[END_REF], we obtain in section 4 results concerning the Orlicz spaces a differentiation basis of rectangles associated to a given sequence of angles, does not differentiate. Combining the results in those two sections with a lemma by A. Stokolos (see Lemma 2 below), we manage in section 5 to prove the three versions of the Theorem 1 stated above.

Notations and Definitions

1.1. Rectangles. For our purposes, a standard interval in R 2 is an interval of the form

Q = [0, L] × [0, ]. We then let Q + := [L/2, L] × [0, ].
Given another (not necessarily standard) interval Q , we shall say that Q has the same shape as Q in case there exists a ∈ R 2 and α > 0 such that Q = a + αQ -calling shape of an interval the quotient of its horizontal side by its vertical one, this is equivalent to say their shapes are equal. Given a family of standard intervals Q, we then denote by B(Q) the family of all intervals in R 2 having the same shape as some Q ∈ Q.

For θ ∈ [0, 2π) we also denote by r θ the (counterclockwise) rotation of angle θ around the origin. Given a set θ ⊆ [0, 2π), we then denote by B θ (Q) the set of rectangles of the form r θ Q for some Q ∈ B(Q).

Differentiation bases.

A family B = x∈R 2 B(x) of measurable subsets of R 2 with positive measure is called a differentiation basis in case the following conditions hold:

(a) for each x ∈ R 2 and each B ∈ B(x), one has x ∈ B; (b) for each x ∈ R 2 , one has inf{diam(B) : B ∈ B(x)} = 0. It is moreover called translation invariant in case one has B(x) = x + B(0) for all x ∈ R 2 , and homothecy-invariant if it is translation-invariant and if moreover αB(0) = B(0) for all α > 0. We shall also say, finally, that the basis B enjoys the Buseman-Feller properties in case for any B ∈ B, the two properties B ∈ B(x) and x ∈ B are equivalent -from now on, we shall see the collections B(Q) and B θ (Q) defined above as differentiation bases enjoying the Buseman-Feller property.

Associated to a differentiation basis B, there is a maximal operator M B defined by:

M B f (x) := sup B∈B(x) 1 |B| B |f |.
When θ ⊆ [0, 2π) is a set of angles, and when B is the collection R θ of all rectangles in the plane, one side of which makes an angle θ ∈ θ with the horizontal line, we shall briefly denote M B by M θ .

One also says that a differentiation basis differentiates a function space X ⊆ L 1 loc (R 2 ) in case for every f ∈ X, the equality:

f (x) = lim diam B→0 B∈B(x) 1 |B| B |f |, holds for a.e. x ∈ R 2 .
1.3. Orlicz spaces. For our purposes, an Orlicz function is a convex, continuous and increasing function Φ : [0, ∞) → [0, ∞) satisfying Φ(0) = 0 and Φ(t) → ∞ at ∞; we say that an Orlicz function Φ satisfies the ∆ 2 condition in case there is an absolute constant K > 0 such that one has Φ(2t) KΦ(t) for all sufficiently large t. The Orlicz function Ψ : [0, ∞) → [0, ∞) defined by Ψ(s) := sup{t|s| -Φ(t) : 0 t < ∞} is then called the complementary function to Φ (a general theory of Orlicz spaces is presented the two monographies by M. A. Krasnosel'skii and Ya. B. Rutickii [START_REF] Krasnosel'skii | Convex functions and Orlicz spaces[END_REF] and by M. M. Rao and Z. D. Ren [START_REF] Rao | Theory of Orlicz Spaces[END_REF]).

Given an Orlicz function Φ, we let L Φ (R 2 ) denote the set of all measurable functions f in R 2 for which Φ(|f |) is integrable (for Φ(t) = t p , 1 p < ∞ this yields the usual Lebesgue space L p (R 2 ), while for Φ(t) = Φ β (t) := t(1 + log + β t), with 0 < β, and for Φ(t) := t(1

+ log + log + t)
we get the Orlicz spaces L log β L(R 2 ) and L log log L(R 2 ), respectively). Given an Orlicz function Φ, recall that a sublinear operator T is said to be of weak type (Φ, Φ) in case there exists a constant C > 0 such that, for all f ∈ L Φ (R 2 ) and all α > 0, one has:

|{x ∈ R 2 : T f (x) > α}| R 2 Φ |f | α .
Whenever Φ(t) = t p for p 1, we shall say that T has weak type (p, p). It is a fact that, for a Buseman-Feller homothecy-invariant differentiation basis B, the following two properties are equivalent for any given Orlicz function Φ:

(i) M B is of weak type (Φ, Φ); (ii) B differentiates L Φ (R 2 ).
The interested reader will find the details of the latter equivalence in [4, Chapter III] (see in particular Remark 4, p. 90). Finally, given an Orlicz function Φ, we shall say that a differentiation basis B differentiates exactly

L Φ (R 2 ) in cases it differentiates L Φ (R 2 ) but fails to differentiate L Ψ (R 2 ) for any Orlicz function Ψ satisfying Ψ(t) = o(Φ(t)) at ∞.

1.4.

A lemma by A. Stokolos. The following useful lemma, which is a particular case of [15, Lemma A], will be useful to us in section 5.

Lemma 2. Assume that Φ is an Orlicz function satisfying the ∆ 2 condition and let R = k∈N R k where, for each k ∈ N, R k is a finite collection of rectangles in R 2 . Assume also that there exists a sequence (λ k ) increasing to ∞ as k → ∞, and a sequence of balls B k satisfying the following properties:

(i) all members of R k have equal area; (ii) for any finite collection S ⊆ R k , one has:

R 2 Ψ R∈S χ R c 1 R∈S |R|,
where Ψ denotes the complementary function to Φ;

(iii) for any R ∈ R k , one has: |R ∩ B k | |R| c 2 λ n ; (iv) | ∪ R k | c 3 Φ(λ k )|E k |;
here, c 1 > 0, c 2 > 0 and c 3 > 0 are constants. Under those assumptions, there exists a differentiation basis B satisfying B ⊆ B(R) that differentiates precisely L Φ (R 2 ).

Let us now move on to the exposition of our results. Before to discuss the type of statements contained in Theorem 1, let us first formulate our observation concerning the boundedness in L p , 1 < p ∞, of the maximal operator M θ .

2. Boundedness of the directional maximal operator in L p and L ∞ spaces We announced the forthcoming proposition in the introduction; the new thing here is that one can include p = ∞ in its statement. Proposition 3. Let θ be as above and let M θ be the directional maximal operator associated to θ defined above. Then the following dichotomy holds:

(1) Either M θ is bounded on L p (R 2 ) for all 1 < p < ∞, in which case the associated differentiation basis B differentiates all L p (R 2 ), 1 < p ∞; (2) or M θ is unbounded on all L p (R 2 ), 1 < p < ∞; in this case the associated differentiation basis B fails to differentiate L p (R 2 ) for all 1 < p ∞.
Proof. Assuming that M θ is unbounded on some L p (R 2 ), 1 < p < ∞, it follows from Bateman [1, Theorem 1, p. 56] that θ admits Kakeya sets, i.e. that for each N ∈ N * there exists a collection R N ⊆ R θ verifying:

(1)

| ∪ R N | 1 N R∈R N R * ,
where one denoted by R * the rectangle having the same center and width as R but three times its length (we hence assume without loss of generality that none of the R's is a square).

We show that M θ is unbounded on all L p (R 2 ) by contradiction. Assume thus that there exists some 1 < q < ∞ such that M θ is bounded on L q (R 2 ). Since R θ is a Busemann-Feller differentiation basis invariant with respect to similarities (translations and homothecies), we know, according to e.g. de Guzmán [4, p. 90], that R θ would then differentiate L q (R 2 ), and hence also L ∞ (R 2 ). It would then follow from de Guzmán [4, Theorem 1.2, p. 69] that there exists a constant c > 0 such that for any bounded measurable set A ⊆ R 2 , one has:

(2)

M θ χ A > 1 4 c|A|.
Define, for N ∈ N * , A N := ∪R N . Observe that for any x ∈ B N := R∈R N R * , there exists some

R x ∈ R N ⊆ R θ such that one has x ∈ R * x . Now compute: M θ χ A N (x) 1 |R * x | R * x χA N 1 |R * x | R * x χR x = |R x ∩ R * x | |R * x | = 1 3 > 1 4 .
We hence have B N ⊆ M θ χ A N > 1 4 ; using (1) and ( 2) we then get:

N |A N | |B N | c|A N |,
which is a contradiction for sufficiently large N . We conclude that M θ is unbounded on all L q (R 2 ), 1 < q < ∞, and that R θ fails to differentiate L ∞ (R n ).

We now proceed towards the proof of Theorem 1 by starting with some geometrical preliminaries that were announced in the introduction.

Some geometrical preliminaries

The following straightforward geometrical fact is borrowed from [START_REF] Moonens | Differentiating along rectangles, in lacunary directions[END_REF].

Lemma 4. Fix real numbers 0 ϑ < θ < π 2 and 0 < 2 < L and let

Q := [0, L] × [0, ]. If moreover one has tan(θ -ϑ) 1/ 1 4 L 2 -1, then r ϑ Q + and r θ Q + are disjoint. Lemma 5. Let {θ j } ⊂ [0, π 4
] be a decreasing sequence such that, letting m j := tan θ j , we have that there exist a constant C > 0, a constant 0 < ζ < 1 and a sequence {t k } so that, for each k, for integers 0 j < k, m j -m k Cζ t k .

Then, there exist constants d(C) > c(C) > 0 and e(C) > 0 depending only on C such that, for each ε > 0 and each integer k

∈ N = N \ {0}, one can find a standard interval Q k = [0, L k ] × [0, k ] and a subset θ k = (θ 0 , . . . , θ k ) ⊂ θ satisfying #θ k = k + 1
and such that the following hold:

(i) 0 < 2 k < L k ε; (ii) c(C)ζ -t 2k L k k d(C)ζ -t 2k ; (iii) θ∈θ k r θ Q k k 2 |Q k |. (iv) for any subset θ = (θ i 0 , . . . , θ i l ) ⊆ θ k (0 l k, i 0 < i 1 < • • • < i l )
and any nonnegative, Borel function ϕ : R + → R + satisfying ϕ(0) = 0, one has:

R 2 ϕ   θ∈θ χ r θ Q k   e(C)|Q k |ζ t 2k l j=0 ϕ(j + 1) l r=j ζ -t ir .
Proof. To prove this lemma, observe first that letting m j := tan θ j for all j 0, we have m j 1 for all j, so that one can compute, for integers 0 j < k:

tan(θ j -θ k ) = m j -m k 1 + m j m k 1 2 (m j -m k ) C 2 ζ t k .
Now choose, for all k, real numbers 0 < 2l < L ε (we write L and instead of L k and k here, for the index k remains constant all through the proof) satisfying:

L 2 = 4 + 16C -2 ζ -2t 2k .
It is clear that one has:

L = 2ζ -t 2k ζ 2t 2k + 4C -2 ,
so that (i) and (ii) hold if we take, for example, c(C) := 2

√ 4C -2 = 4 C and d(C) := 2 √ 1 + 4C -2 . In order to show (iii), define Q := [0, L] × [0,
] and observe that one has: for all integers j < k with k ∈ N . According to Lemma 4, this ensures that the family {r θ j Q + : j ∈ N, 0 j k} consists of pairwise disjoints sets; in particular we get:

tan(θ j -θ k ) C 2 ζ t 2k = C 2 2 C 1 1 4 L 2 -1 = 1 1 4 L 2 -1 , Figure 1. Intersection of rectangles r αr i Q k , 0 i p.
k j=0 r θ j Q k j=0 r θ j Q + = k |Q| 2 ,
(we used to indicate a disjoint union), which proves (iii).

In order to prove (iv), start by writing α r := θ ir in order to alleviate notations in the sequel. Given a finite sequence r 0 < • • • < r p = (p l), observe first that one has (see Figure 1)

p i=0 r αr i Q k = r αr 0 Q k ∩ r αr p Q k ,
from which it follows that one has (see Figure 2):

p i=0 r αr i Q k = 1 2 |OP |h 1 2 k sin P • k 2 k 1 tan(α r 0 -α rp ) , since one has sin P = sin(α r 0 -α rp ) = tan (α r 0 -α rp ) cos (α r 0 -α rp ) 1 2 tan(α r 0 -α rp ) (recall that α r 0 -α rp π/4 π/3. Now let χ := θ∈θ χ r θ Q k and fix 0 j l. Claim 1. One has |{χ = j + 1}| 1 2 2 k l-j s=0 1 tan(αs-α s+j ) .
Proof of the claim. If x ∈ R 2 satisfies χ(x) = j + 1, there exists j + 1 angles α r 0 , . . . , α r j (r 0 < r 1 < • • • < r j ) such that one has

x ∈ j i=0 r αr i Q k = r αr 0 Q k ∩ r αr j Q k = r j r=r 0 r αr Q k .
Hence if r 0 < r 1 < • • • < r j is not a sequence of successive integers, we have in fact χ(x) 1 + r j -r 0 > j + 1. This proves that one has:

{χ = j + 1} ⊆ l-j s=0 s+j r=s r αr Q k = l-j s=0 (r αs Q k ∩ r α s+j Q k ).
Hence one computes:

|{χ = j + 1}| 2 k l-j s=0 1 tan(α s -α s+j ) ,
which proves the claim. Now reminding that one has tan(α s -α s+j ) = tan(θ is -θ i s+j ) C 2 ζ t i s+j for all 0 s l -j, we compute:

|{χ = j + 1}| 2 C k L k k L k l-j s=0 ζ -t i s+j 2 Cc(C) |Q k |ζ t 2k l r=j ζ -t ir .
Letting e(C) := 2 Cc(C) , we finally obtain:

R 2 ϕ(χ) = l j=0 ϕ(j + 1)|{χ = j + 1}| e(C)|Q k |ζ t 2k l j=0 ϕ(j + 1) l r=j ζ -t ir ,
which proves (iv) and hence finishes the proof of the lemma.

The next proposition will be useful in order to study the maximal operator M r θ R .

Proposition 6. Assume that θ is as in Lemma 5. There exists a family Q = {Q k : k ∈ N} of standard intervals in R 2 which is totally ordered by inclusion, verifies inf{diam Q : Q ∈ Q} = 0 and satisfies the following property: for any sufficiently large integer k, there exist sets θ k ⊆ θ and Θ k ⊆ R 2 satisfying #θ k = k + 1 as well as the following conditions (we define

R k := {r θ Q k : θ ∈ θ k } and Y k := ∪R k ): (i) |Y k | γ(C)kζ -t 2k |Θ k |; (ii) for all R ∈ R k , one has: |R ∩ Θ k | |R| γ (C)ζ t 2k ;
(iii) all rectangles in R k have the same area;

(iv) for any subset θ = (θ i 0 , . . . ,

θ i l ) ⊆ θ k (0 l k, 0 i 0 < i 1 < • • • < i l k
) and any nonnegative, Borel function ϕ : R + → R + satisfying ϕ(0) = 0, one has:

R 2 ϕ   θ∈θ χ r θ Q k   γ (C)|Q k |ζ t 2k l j=0 ϕ(j + 1) l r=j ζ -t ir .
where γ(C) > 0, γ (C) > 0 and γ (C) > 0 are constants depending only on C.

Proof. Define R = {Q k : k ∈ N * } where the sequence (Q k ) k∈N * is defined inductively as follows. We choose Q 1 = [0, L 1 ] × [0, 1 ] and θ 1 ⊆ θ associated to k = 1 and ε = 1 according to Lemma 5. Assuming that Q 1 , . . . , Q k have been constructed, for some integer k ∈ N * , we choose Q k+1 = [0, L k+1 ] × [0, k+1 ] and θ k+1 associated to k + 1 and ε = min( k , 1/k) according to Lemma 5. Since the sequence (Q k ) k∈N * is a nonincreasing sequence of rectangles, it is clear that Q is totally ordered by inclusion. It is also clear by construction that one has inf{diam Q k : k ∈ N} = 0. Now fix k ∈ N * and define Θ k := B(0, k ) and R k := {r θ Q k : θ ∈ θ k }. Compute hence, using [Lemma 5, (ii) and (iii)]:

|Y k | k 2 |Q k | = k 2 L k l k = 1 2π k L k l k |Θ k | 1 2π kc(C)ζ -t 2k |Θ k |, so that (i) is proved with γ(C) := c(C) 2π . For any R ∈ R k , there exists θ ∈ θ k such that one has R = r θ Q k ; we hence compute |R ∩ Θ k | |R| = |Θ k ∩ r θ Q k | |Q k | = 1 4 • π 2 k L k k = π 4 • k L k π 4 1 d(C) ζ t 2k ,
which finishes the proof of (ii) if we set γ (C) := π Remark 7. In the conditions of the previous proposition, statement (ii) can be reformulated as follows:

(ii') for all x ∈ Y k , one has

M B θ (Q) χ Θ k (x) γ (C)ζ t 2k .
Using the previous proposition, we can, using standard techniques developed e.g. in a previous work by the second and third authors [START_REF] Moonens | Moving averages in the plane[END_REF] or in a paper by the current authors [START_REF] D'aniello | Averaging on n-dimensional rectangles[END_REF], obtain negative differentiation results in a range of Orlicz spaces for some differentiation bases of rectangles associated to various sets θ.

Bad Orlicz spaces for some maximal functions

In the following statement, we let Φ β (t) := t(1 + log β + t). Proposition 8. Assume that θ is as in Lemma 5, that the sequence t k tends to +∞ as k → ∞ and the sequence { t k β k } is bounded above for some β > 0, that is

M := lim sup k t k β k < ∞.
There exists a (countable) family Q of standard intervals in R 2 with inf{diam Q : Q ∈ R} = 0, satisfying the following conditions:

(i) M B(Q) has weak type (1, 1), and hence the differentiation basis

B(Q) differentiates L 1 (R 2 ); (ii) for any Orlicz function Ψ satisfying Ψ = o(Φ β ) at ∞, M B θ (Q)
fails to be of weak type (Ψ, Ψ), and hence the associated differentiation basis B θ fails to differentiate L Ψ (R 2 ).

Proof. We keep the notations of Proposition 6 and call Q the family of rectangles given by Proposition 6. Observe first that, since Q is totally ordered by inclusion, it follows e.g. from [16, Claim 1] that M B(Q) satisfies a weak (1, 1) inequality.

In order to show (ii), define, for k sufficiently large,

f k := 1 γ (C) • ζ -t 2k χ Θ k ,
where Θ k and Y k are associated to k and Q according to Proposition 6.

Claim 2. For each sufficiently large k, we have:

|{x ∈ R 2 : M B θ (Q) f k (x) 1}| γ 1 (β, C, ζ, M ) R 2 Φ β (f k ).
Proof of the claim. To prove this claim, one observes that for x ∈ Y k we have M B θ (Q) f k (x) 1 according to (ii') in Remark 7. Yet, on the other hand, one computes, for k sufficiently large:

R 2 Φ β (f k ) 1 γ (C) • ζ -t 2k |Θ k | 2t 2k log 1 ζ β 1 γ(C)γ (C) |Y k |(2k) -1 2t 2k log 1 ζ β < 2 β+1 M 1 γ(C)γ (C) |Y k | log 1 ζ β ,
where M := max{1, M } and

γ 1 (β, C, ζ, M ) := log 1 ζ -β γ (C)γ(C) 2 β+1 M .
The claim follows.

Claim 3. For any Φ satisfying Φ = o(Φ β ) at ∞ and for each constant T > 0, we have:

lim k→∞ R 2 Φ β (|f k |) R 2 Φ(T |f k |) = ∞.
Proof of the claim. Compute for any k:

R 2 Φ(T |f k |) R 2 Φ β (|f k |) = Φ(ζ -t 2k T /γ (C)) Φ β (ζ -t 2k /γ (C)) = Φ(ζ -t 2k T /γ (C)) Φ β (ζ -t 2k T /γ (C)) Φ β (ζ -t 2k T /γ (C)) Φ β (ζ -t 2k /γ (C)) ,
observe that the quotient

Φ β (ζ -t 2k T /γ (C)) Φ β (ζ -t 2k /γ (C))
is bounded as k → ∞ by a constant independent of k, while by assumption the quotient Φ(ζ -t 2k T /γ (C))

Φ β (ζ -t 2k T /γ (C)) tends to zero as k → ∞. The claim is proved.

We now finish the proof of Proposition 8. To this purpose, fix Φ an Orlicz function satisfying Φ = o(Φ β ) at ∞ and assume that there exists a constant T > 0 such that, for any α > 0, one has:

|{x ∈ R 2 : M B θ (Q) f (x) > α}| R 2 Φ T |f | α .
Using Claim 2, we would then get, for each k sufficiently large:

0 < γ 1 (β, C, ζ, M ) R 2 Φ β (f k ) x ∈ R 2 : M B θ (Q) f k (x) > 1 2 R n Φ(2T f k ),
contradicting the previous claim and proving the theorem.

The following result is proved in a very similar way -reason for which we here omit its straightforward proof. Proposition 9. Assume that θ is as in Lemma 5, that the sequence t k tends to +∞ as k → ∞, that it satisfies M := lim sup k log t k k < ∞ and define Φ(t) = t(1 + log + log + t). There exists a (countable) family Q of standard intervals in R 2 with inf{diam Q : Q ∈ Q} = 0, satisfying the following conditions:

(i) M B(Q) has weak type (1, 1), and hence the associated differentiation basis B(Q) differentiates L 1 (R 2 ); (ii) for any Orlicz function Ψ satisfying Ψ = o(Φ) at ∞, M B θ (Q) fails to be of weak type (Ψ, Ψ) -hence the differentiation basis B θ (Q) fails to differentiate L log log L(R 2 ).

We now turn to proving the three versions of Theorem 1 we stated in the introduction.

Three examples of sequences yielding Theorem 1

Let d ∈ N be fixed. Assume that the sequence (θ k ) k∈N ⊆ (0, π/4] is such that one has:

(3)

0 < λ < lim j→∞ θ j+1 θ j d lim j→∞ θ j+1 θ j d < µ < 1.
Letting m j := tan θ j for all j ∈ N, one clearly has:

lim j→∞ m j θ j = 1,
so that (3) also holds for the sequence (m j ) j∈N . There hence exists an index j 0 ∈ N such that, for all j j 0 , one has λ

m j+1 m j d
µ (we may also and will assume that one has λ 2 m j 0 λ). For the sake of clarity, we shall now consider that j 0 = 0 and compute, for an integer 0 j < k:

tan(θ j -θ k ) = m j -m k 1 + m j m k 1 2 (m j -m k ).
We also have, for every integer 0 j < k:

(4) λ k-j-1 i=0 d i m j d k-j m k µ k-j-1 i=0 d i m j d k-j .
5.1. Assume first that d = 1. Arguing as in [START_REF] Moonens | Differentiating along rectangles, in lacunary directions[END_REF], we then get:

tan(θ j -θ k ) 1 2 m 0 • λ k • [µ -1 -1],
so that the if we set C = 1 2 m 0 • [µ -1 -1], ζ = λ, t k = k and take β = 1 then the hypotheses of Proposition 8 are satisfied and let Q, R k and θ k be associated to θ by Proposition 6; define Φ(t) := t(1 + log + t) and observe that it is easy to see (see e.g. [9, Chapter 1]) that one can have Ψ(t)

K 1 e t for the complementary function Ψ to Φ. There is no loss of generality, of course, to assume ζ < 1/e. Now fix a subset θ = {θ ir : 0 r l} ⊂ θ k (0 i 0 < i 1 < • • • < i l k) and write, using ϕ = Ψ in [Proposition 6, (iv)]:

R 2 Ψ   θ∈θ χ r θ Q k   K 1 |Q k |ζ 2k l j=0 e j l r=j ζ -ir ,
where K 1 = eK 1 e(C) > 0. Write then:

l r=j ζ -ir ζ -i l ∞ s=0 ζ s ζ -k 1 -ζ .
On the other hand one computes: Hence we obtain:

R 2 Ψ   θ∈θ χ r θ Q k   K 1 |Q k |ζ 2k • e k+1 e -1 • ζ -k 1 -ζ 2K 1 1 -ζ |Q k |,
since one has eζ < 1 and e/(e -1) 2.

Combining now Proposition 8 and Lemma 2 for the particular sequence considered in this section, we get the following result.

Theorem 10. Assume that the sequence (θ k ) k∈N ⊆ (0, π/4] is such that one has:

0 < λ < lim j→∞ θ j+1 θ j lim j→∞ θ j+1 θ j < µ < 1.
Under those assumptions, there exists a countable family of standard intervals in R 2 denoted by Q and totally ordered by inclusion, satisfying the following properties (where one defines Φ(t) := t(1 + log + t)):

(i) the associated differentiation basis B(Q) differentiates L 1 (R 2 ); (ii) for any Orlicz function Ψ satisfying Ψ = o(Φ) at ∞, the differentiation basis B θ (Q) fails to differentiate L Ψ (R 2 ); (iii) there exists a differentiation basis B ⊆ B θ (Q) that differentiates exactly L log L(R 2 ).
Remark 11. Statement (iii) above is weaker than stating that the basis B θ (Q) itself does differentiate (exactly) L log L(R 2 ); however, it is not known to us whether this statement holds or not. 5.2. Assume now that d > 1 in (3). Using (4), we obtain for 0 j < k:

λ d k-j -1 d-1 m j d k-j = λ k-j-1 i=0 d i m j d k-j m k µ k-j-1 i=0 d i m j d k-j = µ d k-j -1 d-1 m j d k-j .
We then compute, for the same j < k: Fix as before a subset θ = {θ ir : 0 r l} ⊂ θ k (0 i 0 < i 1 < • • • < i l k) and write, using ϕ = Ψ in [Proposition 6, (iv)]:

tan(θ j -θ k ) 1 2 (m j -m k ) 1 2 [m k 1 d k-j • µ - k-j-1 i=0 d i d k-j -m k ] 1 2 m k • [µ - k-j-1 i=0 d i d k-j -1] 1 2 m 0 d k • λ k-1 i=0 d i • [µ -1 d-1 -1] = 1 2 m 0 d k • λ d k • [µ -1 d-1 -1] 1 2 m 0 d k • λ d k • [µ -1 d-1 -1] 1 2 ( λ 2 ) d k • λ d k • [µ -1 d-1 -1] 1 2 ( λ 2 ) 2d k • [µ -1 d-1 -1]. So if we set C = 1 2 [µ -1 d-1 -1], ζ = ( λ 2 ) 2 and t k = d k ,
R 2 Ψ   θ∈θ χ r θ Q k   K 2 |Q k |ζ d 2k l j=0 e e j+1 l r=j ζ -d ir ,
where K 2 = K 2 e(C) > 0. Write then:

l r=j ζ -d ir ζ -d i l ∞ s=0 ζ s ζ -d k 1 -ζ .
On the other hand one computes: Hence we obtain:

R 2 Ψ   θ∈θ χ r θ Q k   K 2 e|Q k |ζ d 2k • e e k+1 e -1 • ζ -d k 1 -ζ K 2 e (1 -ζ)(e -1) |Q k | exp e k+1 + d k -d 2 k ,
since one has eζ < 1. Observing e.g. that d 2 4 > e and that one hence has:

e k+1 + d k -d 2 k = e e k - 1 2e d 2 k + d k - 1 2 d 2 k ,
and since moreover it is clear that a k -εb k tends to -∞ as k → ∞ for any real numbers 1 < a < b and ε > 0, we finally get:

lim k→∞ exp e k+1 + d k -d 2 k = 0.
There thus exists a constant K 2 > 0 (depending only on θ and d) for which one has:

R 2 Ψ   θ∈θ χ r θ Q k   K 2 |Q k |.
Combining now as above Proposition 9 and Lemma 2 for the particular case of a sequence satisfying (3) for d > 1, we obtain the following result.

Theorem 12. Assume that d ∈ N * and the sequence (θ k ) k∈N ⊆ (0, π/4] are such that one has:

0 < λ < lim j→∞ θ j+1 θ d j lim j→∞ θ j+1 θ d j < µ < 1.
Under those assumptions, there exists a countable family of standard intervals in R 2 denoted by Q and totally ordered by inclusion, satisfying the following properties (where one defines Φ(t) := t(1 + log + log + t)):

(i) the associated differentiation basis B(Q) differentiates L 1 (R 2 ); (ii) for any Orlicz function Ψ satisfying Ψ = o(Φ) at ∞, the differentiation basis B θ (Q) fails to differentiate L Ψ (R 2
); (iii) there exists a differentiation basis B ⊆ B θ (Q) that differentiates exactly L log log L(R 2 ).

A non-lacunary example.

Let (a j ) be a nonincreasing sequence of positive real numbers with 0 < a = inf j a j sup j a j = b < 1. Clearly, a 0 = max a j = b. In the sequel we fix a real number 0 < d < 1 and we define a sequence (θ j ) by: θ j := arctan (a j ) j d .

Observe that letting m j := tan θ j for all j, we can write, for j 0 j k:

m j -m k = (a j ) j d -(a k ) k d (a k ) j d -(a k ) k d = (a k ) k d (a k ) j d -k d -1 a k d 1 b k d -j d -1 a k d 1 b k d -(k-1) d -1 . (5)
It hence follows in particular that (m j ) (and hence also (θ j )) is a decreasing sequence.

On the other hand, it is easy to observe that one has, for all j:

j d -(j -1) d = dj d-1 + O(j d-2 ),
we get:

j d -(j -1) d j 1-d = d + O(j -1 ),
so that, for j sufficiently large, we have:

(6) j d -(j -1) d d 2j 1-d .
It is also the case that for j sufficiently large, we always have:

(7) 2 d j 1-d 1 2 b -j d ;
we shall hence assume that both ( 6) and ( 7) hold for j j 0 , and we shall, from now on, work with the sequence (θ j ) j j 0 ; we also define θ := {θ j : j j 0 }. Now given k > j 0 , we obtain from ( 5), for all max{j 0 , log a (log a e)} j 1 j k , using the inequality a x 1 + x for all x log a (log a e) and then ( 6) and ( 7): Hence we obtain:

m j -m k a k d 1 b k d -(k-1) d -1 a k d d 2k
R 2 Ψ   θ∈θ χ r θ Q k   K 3 |Q k |(k + 1) 2 ζ (2k) d -k d e k d .
Yet one has:

(k + 1) 2 ζ (2k) d -k d e k d = (k + 1) 2 eζ 2 d -1 k d = (k + 1) 2 η k d → 0, k → ∞.
Hence there exists a constant K 3 > 0, independent of k, for which one has:

R 2 Ψ   θ∈θ χ r θ Q k   K 3 |Q k |.
Combining now Proposition 8 and Lemma 2 for the particular sequence considered in this section, we get the following result.

Theorem 13. Assume that 0 < d < 1 is a real number. Let (a j ) be a nonincreasing sequence of positive real numbers with 0 < inf j a j sup j a j < 1 and define a sequence (θ j ) by: θ j := arctan (a j ) j d .

Under those assumptions, there exists a countable family of standard intervals in R 2 denoted by Q and totally ordered by inclusion, satisfying the following properties (we define Φ(t) := t(1 + log 1/d + t) for β > 0): (i) the associated differentiation basis B(R) differentiates L 1 (R 2 ); (ii) for any Orlicz function Ψ satisfying Ψ = o(Φ) at ∞, the differentiation basis B θ (Q) fails to differentiate L Ψ (R 2 ); (iii) there exists a differentiation basis B ⊆ B θ (Q) that differentiates exactly L log 1 d L(R 2 )hence it also differentiates L p (R 2 ) for all p > 1.

Remark 14. Let (a j ) and d be as above, that is let {a j } be a non-increasing sequence of positive real numbers with 0 < a = inf j a j sup j a j = b < 1 and 0 < d < 1. Then (a j ) j d cannot be lacunary according to the definition in [START_REF] Moonens | Differentiating along rectangles, in lacunary directions[END_REF]. Assume, by contradiction that there exist α and β satisfying 0 < α a j+1 (j+1) d a j j d < β < 1.

Then, for each j 2, it is , so that it must be lim j a j = 0, and this contradicts the hypothesis inf j a j = lim j a j = a > 0.

  ii) the differentiation basis B θ obtained from B by allowing its elements to rotate of an angle θ ∈ θ around their lower left vertex, fails to differentiate L Ψ (R 2 ) for any Orlicz function Ψ satisfying Ψ = o(Φ) at ∞; (iii) there exists a differentiation basis B ⊆ B θ which differentiates L Φ (R 2 ) but fails to differentiate L Ψ (R 2 ) for any Orlicz function Ψ satisfying Ψ = o(Φ) at ∞ (we then say that B differentiates precisely L Φ (R 2 )).

Figure 2 .

 2 Figure 2. Intersection of rectangles r αr i Q k , 0 i p.

4 1d

 4 (C) . Now (iii) is clear, while (iv) results immediately from [Lemma 5, (iv)].

l j=0 e e j+1 e e l+1 ∞ s=0 e -s e 1+e k+1 e - 1 .

 s=01

where K 3 =-k d , since we have ζ i d l -i d r 1

 31 1-d 2(ab) k d . So if we take C = 2, ζ = ab, t k = k d and β = 1 d , we have that the hypotheses of Proposition 8 are satisfied. Let also Q, R k and θ k be associated to θ by Proposition 6; define now Φ(t) := t(1 + log 1/d + t) and observe that it is easy to see (see e.g. [9, Chapter 1]) that one can have Ψ(t) K 3 e t d for the complementary function Ψ to Φ. There is no loss of generality, reducing ζ if necessary, to assume that one has η:= eζ 2 d -1 < 1. Now fix a subset θ = {θ ir : 0 r l} ⊂ θ k (0 i 0 < i 1 < • • • < i l k) and write, using ϕ = Ψ in [Proposition 6, (iv)]: K 3 e(C) > 0. Write then: for all j r l. On the other hand one computes in a similar fashion: l j=0 e j d = e l d l j=0 e j d -l d (k + 1)e k d .

  then the hypotheses of Proposition 9 are satisfied. Let hence Q, R k and θ k be associated to θ by Proposition 6 and define Φ(t) := t(1 + log + log + t) and observe that it is easy to see (see e.g. [9,Chapter 1]) that one can have Ψ(t) K 2 exp(exp t)for the complementary function Ψ to Φ. There is no loss of generality, again, to assume ζ < 1/e.

Acknowledgements. Both authors would like to thank their respective research institutes for the warm hospitality they could enjoy from them when preparing the present manuscript. The second author would like to acknowledge the support of the "Laboratory Ypatia of Mathematical Sciences", thanks to which the current collaboration has been possible.