Anne Gebhardt

Soil micromorphological data from traditional and experimental agriculture

come

Introduction

Recent soil micromorphological investigations on palaeosols buried under well dated archaeological structures, specific pit and ditch infills, and colluviums in Brittany, show that the present landscape, developed in Brittany since the last glacial period, has been intensively affected by human activity since the Neolithic (Gebhardt 1990[START_REF] Gebhardt | Micromorphological evidence of soil deterioration since the mid-Holocene at archaeological sites in Brittany, France[END_REF]. Micromorphological pedofeatures have provided much data on soil evolution in Brittany (palaeopedogenesis, degree of erosion) and help us to understand the way a site was affected by human interference (deforestation, agriculture, etc). Over and above the general tendency of some profiles to acidify under human impact, different types of nonrcontemporaneous edaphic evolution can be distinguished. There are typical examples of soil degradation initiated by human clearance as early as the Neolithic. In some cases of profile rejuvenation, probably caused by the general erosion of the loessic blanket, only a slight soil degradation has been observed. Deforestation also instigated the rise of watcr-tables, causing soil hydromorphism. The micromorphological study of Brittany's palaeosols demonstrates the complexity of ancient pedological phenomena which occur in soils. However, this interesting problem encourages research, especially the study of Holocene palaeosols. The analysis of some colluvial deposits showed that although commencing with the first deforestations, colluviation was probably further encouraged by agricultural activity and the loamy character of the soils. The variation in thickness observed in the downslope deposits may also be related to the duration of cultivation. In receiving features, ditches, lynchets etc, colluviated material can become trapped and develop pedologically in buried but poorly sealed conditions. In the example studied, comparison between ditches may reveal differences in the microstructure of the infilling, even on the same site and under the same modern agricultural treatment. In that case, specific pedofeatures may be related to the different function of each ditch and the evolution of its fill. As a result even ditch maintenance practices could be determined. Until now, results from palaeosol studies have provided more data on anthropogenic deforestation, landuse management over a long period and the influence of early agriculture on the environment (soil acidification, erosion) than on the types of agricultural implements used in the past and how they affected the soil. In the case of unprotected soils and infill sediments, the interpretation of micro-pedofeatures is more difficult than for buried palaeosols. Possibly more care must be taken in their interpretation, especially in modern, heavily cultivated areas. Here, more than ever, micromorphological data should be used within a multidisciplinary framework in order to compare data from other investigative methods (charcoal, seeds or pollen analysis). Because the preservation of such organic residues can be very poor in some archaeological contexts, it is important to compare micromorphological data from archaeological structures with an experimental and traditional data base when attempting to explain the function of structures. This may be the only way to classify the pedofeatures formed Within infills dating to the use of the structure, and those formed after its use, as for example by subsequent 1ong-term agriculture.

Why conduct experiments in archaeological soils?

Although there is useful modern agricultural research [START_REF] Barratt | Micromorphological observations on the effects ofland use differences on some New Zealand soils[END_REF]Collins and Larney 1987;[START_REF] Dexter | Prediction of soil structure produced by tillage[END_REF][START_REF] Jongerius | The role ofmicromorphology in agricultural research[END_REF][START_REF] Kooistra | The effects of compaction and deep tillage on soil structure in dutch sandy loam soil[END_REF][START_REF] Pagliai | Effects of different managements practices on soil structure and surface crusting[END_REF] which can be used as analogues (Macphail et al.1990), there are few experimental data on the impact of ancient implements and ancient field management methods on the soil microstructure. Micromorphological descriptions of ancient soils are given using the most recent nomenclature [START_REF] Bullock | Handbook for Soil Thin Section Description[END_REF]. Archaeological and environmental interpretations themselves are based on micropedological investigations carried out within a soil science context [START_REF] Barratt | A micromorphological investigation of structural changes in the top soil of Patamahoe clay used for market gardening[END_REF][START_REF] Jongerius | Some morphological aspects of regrouping phenomena in Dutch soils[END_REF][START_REF] Jongerius | The role ofmicromorphology in agricultural research[END_REF][START_REF] Pagliai | Micromorphological study of crusts[END_REF] or deduced from micromorphological observation carried out within an archaeological pluridisciplinary context [START_REF] Courty | Soils and Micromorpholugy in Archaeology[END_REF]). In the case of ancient cultivated fields, only a few micromorphological interpretations have been carried out on the kind of tool used (hoe, ard, plough, spade), and how this affects soil microstructure. At Strathallan, for example, counting of the dusty clay coatings present in the palaeosols permitted [START_REF] Romans | The general effects of agriculture on the soil profile[END_REF] to form the hypothesis that in the upper 6 cm of soil concentration of coatings was due to hoe cultivation at the mound site. At the Henge site, the concentration of coatings at 12 cm was interpreted as resulting from ard cultivation, since an ard ploughs deeper than a hoe. It is not known, however, if this result can be applied to all ancient cultivated sites. The realisation that this hypothesis had to be checked was an important initial impetus for the start of an experimental research programme. Experimental investigations are now strongly linked to the archaeological world (see the Actes of the international symposium Experimentations en archéologie: bilan er perspectives [START_REF] Devos-Firmin | Une fenne gauloise a Parchéodrome de Beaune (Côtes d'Or)[END_REF]), and are as diverse as experiments on architecture [START_REF] Devos-Firmin | Une fenne gauloise a Parchéodrome de Beaune (Côtes d'Or)[END_REF][START_REF] Sainty | Construction experimentale d' une maison Néolithique a Holtzheim (Bas-Rhin)[END_REF][START_REF] Reynolds | Deadstock and livestock[END_REF], lithic material and everyday life conditions [START_REF] Coles | Experimental Archaeology[END_REF]. Experimental agriculture in archaeology has recently appeared in Europe, for example in the study of plant change through domestication [START_REF] Willcox | Archaeological significance of growing Near Eastem progenitors of domestic plants at Jalès (France)[END_REF]. To this end seed collections have been made, ancient crops have been cultivated, and their yields studied [START_REF] Reynolds | Deadstock and livestock[END_REF][START_REF] Luning | Experimenteller Gctreideanbau im Hambacher Forst, Gemeinde Elsdorf[END_REF][START_REF] Anderson | Experimental cultivation, harvest and threshing of wild cereals and their relevance for interpreting the use of Epipalaeolithic and Neolithic artefacts[END_REF]. Experimenters have also been interested in the use of reconstructions of ancient ploughing and harvesting tools, and to study the microwear traces which forrn on them [START_REF] Reynolds | Deadstock and livestock[END_REF][START_REF] Lerche | Ridged fields and profiles of plough-furrows. Ploughing practices in Medieval and Post-Medieval Time[END_REF][START_REF] Anderson | Experimental cultivation, harvest and threshing of wild cereals and their relevance for interpreting the use of Epipalaeolithic and Neolithic artefacts[END_REF]. Because experiments sometimes take a considerable time to produce results, an alternative research avenue is the study of traditional societies and their technologies [START_REF] Leroi-Gourhan | L'Homme er la Matière[END_REF][START_REF] Leroi-Gourhan | L'Homme er la Matière[END_REF][START_REF] Pétrequin | Gens de l'eau, gens de la terre[END_REF][START_REF] Butler | Pulse agronomy: traditional systems and implications for early archaeology[END_REF]). However, in all these investigations, except for a few works on ethno-pedology [START_REF] Wilshusen | An ethno-archaeological perspective on soils[END_REF]) and on experimental furrow shape [START_REF] Lerche | Ridged fields and profiles of plough-furrows. Ploughing practices in Medieval and Post-Medieval Time[END_REF], soils have been largely ignored, and research seldom deals with the impact of tool and agricultural management on the soil.

Some experimental results

ln order to compare the microstructures of archaeological soils with experimental and traditionally cultivated soils, the same sampling and laboratory techniques should be used for both. For methodology, see Macphail and Goldberg this volume;Matthews this volume;Gebhardt 1990[START_REF] Gebhardt | Micromorphological evidence of soil deterioration since the mid-Holocene at archaeological sites in Brittany, France[END_REF][START_REF] Courty | Soils and Micromorpholugy in Archaeology[END_REF]. The results described below come from several sites in north west Europe (Fig. 1). In order to maintain the best pedoclimatic unity for these particular experiments, and also to allow better comparison with archaeological sites studied in Brittany, the experiments are divided roughly into two groups: the experimental sites of Butser (United Kingdom), Hambacher Forst (Germany), Grignon (France), Melrand (France) and the traditional sites of Malguénac (France) and Melrand (France).

Micromorphological analysis of soil structure modification caused by different (ancient) cultivation implements

The aim of this programme (see Gebhardt 1990[START_REF] Gebhardt | Micromorphological analysis of soil structural modification caused by different cultivation implements[END_REF] was to study the modifications in structure and porosity produced in soil by various cultivation implements, as both structure and porosity are important factors in agricultural soils [START_REF] Fedoroff | Un plaidoyer en faveur de la paléopédologie[END_REF]). The structure in soil thin-sections was observed through a stereoscopic microscope.

The four experimental sites and their soil types are as follows:

-Butser Ancient Farm (United Kingdom): a calcareous colluvium (cambisol, F.A.Oi 1975;sol brun: Duchaufour 1983) ploughed (depth: 30 cm) with a reconstructed Iron Age type ard since 1970 [START_REF] Reynolds | Deadstock and livestock[END_REF]; -Hambacher Forst (Germany): an alfisol (F. A.O. 1975;sol bmn lessivé: Duchaufour 1983) developed in loessic-loam under forest, the soil of which was ploughed (5 to 10 cm deep) several times within one year with an ard pulled by two people. Between each of the eight rounds of four criss-cross ard ploughing (N/S, E/W, NW/SE, NE/SW) the field surface was trampled by three to four people for five minutes [START_REF] Meurers-Balke | Experimcntc zum Anbau und Zur Verarbeiterung prahistotorischer Getreidearten[END_REF]Tegmüller pers.comm.). A plot nearby, worked with a motor cultivator (Ap horizon: 10 cm deep) was also studied for companson;

-Grignon Forest (France, INA-PG) : an alfisol (F. A.O. 1975;sol bmn lessivé: Duchaufour 1983) developed on a loessic loam was worked (5 to 10 cm deep) with a wooden hoe of Neolithic type (reconstructed from the Champréveyre type, Switzerland) and heavily watered over a period of a single month in order to accelerate the pedological processes (about 2000 litres in July 1986); The micromorphological impact of four ploughing tools was then compared to uncultivated reference profiles at each site 1 the hoe (Grignon), the ard (Butser, Hambacher Forst), the spade (Dehérain) and the cultivator (Dehérain, Hambacher Forst). For each kind of implement, the degree of soil compaction, porosity, ped form and shape, depth of organic matter burial, and the depth of tool impact which marked the depth of the Ap horizon, were compared (Gebhardt 1990[START_REF] Gebhardt | Micromorphological analysis of soil structural modification caused by different cultivation implements[END_REF]. At Dehérain, Butser and Grignon, the initial structure of the soil was very compacted and well structured, with characteristic 'birdfoot-shaped' cracks (3 or 4 radial junctions) isolating soil peds.

Significant biological activity occurred, suggested by faunal pellets, root remains and its high porosity formed mainly of channels and burrows. The loessic loam at Hambacher Forst is very compact, but less porous than the other sites. After cultivation, the soil at Dehérain, Butser and Grignon became looser, comprising clods of different sizes according to the tool employed (a.rd or spade: mean clod size 1O mm; cultivator: mean clod size 15mm) mixed with smaller angular peds. In the middle of the biggest clods, flle original soil structure can still be preserved. At Hambacher Forst, the soil microstructure collapsed because the original A horizon structure was weak, but numerous rounded peds (mean diameter 2.5 mm) were observed in the groundmass compacted by the a.rd. The bottom of the ploughed horizon is compacted into a lmm thick 'pan' formed during the one year experiment. This initial feature is related in origin to plough pans in modem arable soils [START_REF] Jongerius | Some morphological aspects of regrouping phenomena in Dutch soils[END_REF]Collins and Larney 1987). The motor cultivator created clods with straight, cleanly broken edges. The hoeing experiment at Grignon showed an increase in 'star-shaped' porosity. This was certainly caused by excessive watering of the plots over too short a time [START_REF] Courty | Application dc la micromorphologie à la caractérisation des sols anciens cultivés[END_REF]. Input of organic matter in the form of pure straw, humified straw, or stable litter did not affect the structure of any horizon, worked or not. The only evidence ofthis input was an increasing quantity of organic fragments, more or less incorporated into the soil. Traces of faunal activity seem to be more visible in thin section in the uncultivated control soils and unmanured plots, than in the manured plough horizons. Whatever the treatment, the excrcments of microfauna (oribatids, enchytraeids) were more obvious than mesofauna (earth-worms). Cultivating the soil (ploughing, wetting and drying effects) obscures or destroys most of the micromorphological traces of faunal activity. Pellets and crumbs are broken and/or coalesce into more compactly structured peds. The biological fabric remains visible deeperin the soil profile, beneath the level of ploughing.

A traditional ploughed field (see Gebhardt 1990)

Ethnological survey allowed us to find in Malguénac (Morbihan, France), an ancient field which had been ploughed with a horse-drawn plough, although for the last fifty years the land has been used as a cow pasture. As the profile had not been disturbed since cultivation ceased, it is possible to recognise the typical horizons of a distric cambisol (F. A.O. 1975;brun acide: Duchaufour 1983) developed on weathered granite. The profile has a brown-yellow sandy loam Ap (15 to 20 cm) horizon, including 10 cm long 'comma shaped' fragments of B horizon brought up by plough action. Under the Ap, there is a yellowish sandy-loam B horizon (15 cm thick). The boundary is generally sharp for the 'comma' fragments, sharp between Ap and B, and diffuse between B and the weathered granite.

Under the microscope, we can recognise the following:

-The ploughed Ap horizon includes fragments of the anient leached A1 horizon material (1 .5 mm) and yellowish fragments of B horizon (size: 0.5 to 2 cm, Fig. 2a). Microfaunal activity (enchytraeids pellets) is Well developed in all this horizon. Earthworm channels are rare, concentrated at the top of the profile, and indicate the recent rise in pl-I due to manuring or pasture development;

-At around 25 cm depth, in the B horizon, it is possible to recognize relict fragments of an ancient Bt horizon characterised by some limpid clay void coatings and infillings, which are sometimes layered and/or fragmented. At this level some other void clay coatings are very dusty and unlayered (Fig. 2b). They are more recent than the limpid coatings. In some places the groundmass is also dusty. The dusty deposits can be related to the ancient cultivation of the field [START_REF] Jongerius | Some morphological aspects of regrouping phenomena in Dutch soils[END_REF]. Biological activity is represented by enchytraeid pellets.

There is no obvious structural evidence of ancient ploughing, because the microstructure has probably been destroyed by biological activity since its pastural use.

Other soil micromorphological features relating to experimental agricultural practices and human activities

Lann-Gouh (near Melrand, Morbihan, France) is a medieval (10th century AD) deserted village situated on the edge of a granitic plateau dominating the river Sarre [START_REF] Chalavoux | La ferme archéologiques de Melrand. Dalc'homp sonj[END_REF]. Since 1986, the scientific investigation of this site by J. Chalavoux has been the focus of original multidisciplinary investigations of the medieval site and experiments related to archaeological problems [START_REF] Chalavoux | De la fouille à l'expérimentation en passant par les archéosciences: l'exemple de la micromorphologie[END_REF]).

The pig pasture

On the north-east slope of the Lann-Gouh plateau, pig breeds, genetically close to the ancient 'Pores de Bayeux' breeds, were bred in an open air forested enclosure. Great damage was caused by them in just a few weeks: very quickly, the soil surface was completely disturbed and small trees were uprooted. This provided an opportunity to compare the soil micrornorphological impact of the pig pasture with the untouched forest soil (Gebhardt 1990). The forest soil profile (top 40 cm studied) is a distric cambisol (F. A.O. 1975;brun aeide: Duchaufour 1983) It has a dark-brown very bioturbated upper A0 organic horizon (3 cm deep, Fig. 3a), with a slightly open porosity. The 10 cm deep brown homogenous Al horizon is also rich in weathered organic matter and includes roots. Microfaunal activity (numerous enchytraeid pellets) is intense in both upper horizons. The mineral B horizon is developed on the weathered granite. The junction between horizons is diffuse. The pig pasture profile is essentially the same as the one in the forest, but is buried by a 15cm thick compacted colluvial horizon (Fig. 3b), formed by a mixture of organic horizon (Ao and Al) and yellowish-brown compacted B horizon soil fragments. The boundary between the buried Ao horizon and the colluvium is sharp. The result of pig action was to remove the field surface and to generate a colluvial deposit on the slope.

The threshing area (before and after ase) At St Price in the same area of Lann-Gouh (Melrand, Morbihan), a threshing area was reconstructed in a farm yard with the help of retired farmers who had used traditional construction techniques in their youth (Chalavoux unpubl.). The threshing area was constructed with loamy argillic gley soil collected from a local wetland, first mixed by horse trampling. A Fest Noz (typical breton dancing party) was organised in order to further pack the sediment in a l0 cm thick layer across the 7 m wide threshing area. Rye was threshed with ancient flails in a traditional way: straw bundles being arranged in a thick bed across the area before being threshed by farmers. Grain was collected by scraping the surface with a wooden scraper which would not damage the area's surface. Sampling was carried out on this threshing floor, before and after the threshing, to provide control and post-threshing samples. Micromorphological analysis showed very few differences between the two samples. The microstmcture of the threshing area is compact and the porosity is very low, suggesting greater compaction. Diatoms are numerous, often associated with organic matter coming from the marshy sediment. Phytoliths are also numerous, although the longer phytoliths appear to have been broken. The unlayered sediment shows an alternation of units rich or poor in fine material including some orientation of the plant fragments. This textural fabric reflects the repeated mixing of sediment during the construction of the area. In contrast to what was expected, there is no cracking due to drying or compaction from threshing. The sediment appeared to have dried progressively through its homogeneous thickness. During threshing, the flail, which strongly beats the rye, has no influence on the area's constructed structure, or on the grain (rye grains were neither buried nor broken up during threshing).

The house floor

At Lann-Gouh, different dwelling floors were investigated. Inside a medieval house, pieces of a compacted earth were excavated. It was impossible to know if it was a naturally compacted or manmade earth floor, or to give any details on the way it was constructed.

Through the microscope, we compared this compacted earth with two other examples, gathered in the Lann-Gouh area: -a traditional earth»floor from a house built last century, constructed by spreading successive layers of fine and coarse material; -an experimental house earth floor, which resulted from natural soil being compacted by human and animal trampling.

The three earth floors looked different micromorphologically (Gebhardt l99O). The traditional house floor is mainly mineralogenic and well layered (Fig/ la), whereas the experimental one is organic with pseudo-horizontal cracks (Fig/lb). The archaeological example, in contrast, is very homogeneous, with few cracks and looks like the soil beneath the other two examples (Fig. 4c). It was concluded that the archaeological fragments are in fact from the top of the weathered granitic subsoil.

Thus we can suppose that the inhabitants of the medieval house lived on a natural soil floor, from which the organic upper horizon had been removed. It simply became compacted by domestic trampling.

Discussion

This last example shows how important experimental and traditional data can be in archaeological interpretation. It was possible to give an interpretation on the formation ofhouse floors. which would have been impossible on the basis of archaeological data alone. However, we found that experimental data can he very difficult to use in archaeological interpretations unless some care is taken.

The requirement for pedoclimatic homogeneity in experimental design

In the case of the house floors, comparisons were very precise because the comparative material was gathered in exactly the same pedoclimatic area. Experimental results obtained from the experimental farms on calcareous soils were difficult to compare with archaeological sites studied in Brittany because the latter are located on granitic and metamorphic substrata. For example, tillage micropedofeatures such as the dusty clay coatings described by [START_REF] Romans | The general effects of agriculture on the soil profile[END_REF] were not observed at the four experimental cultivation sites (except for perhaps some concentration of textural feature at Hambacher Forst which will be discussed below). This lack of dusty clay formation is not a question of the length of experiment duration (clay translocation has been rapidly produced in laboratory experiments] see [START_REF] Dalrymple | Experimental construction of illuviation cutans (channel argillans) with differing morphological and optical properties[END_REF]) and the fields at Butser have been ploughed since 1974 [START_REF] Reynolds | Deadstock and livestock[END_REF]. Nor is it a question of quantity of rainfall (see Grignon above; Gebhardt 1992). Clay translocation occurs in decalcified soils [START_REF] Duchaufour | Pédogénese et classification[END_REF]. Most of the experimental profiles were not sufficiently decalcified. At Malguénac, where 25-30 cm deep tillage (plough) had taken place, dusty clay coatings were observed because the profile was sufficiently decalcified. Even if these experimental results were not comparable to archaeological sites in Brittany, they were very useful in interpreting structural break-down by tillage at Ashcombe Bottom in England [START_REF] Macphail | Soil micromorphological evidence of early agriculture in north-west Europe[END_REF][START_REF] Macphail | Soil micromorphological evidence of ancient soil erosion[END_REF].

The experimental site's historical background

At Harnbacher Forst, the presently wooded experimental site was first deforested during the Roman period (Meurers-Balke pers. comm), which would also have led to dusty clay coating formation. Thus, the few thin layers of dusty clay observed at 12-14 cm depth in the voids of the experimental profile which probably relate to experimental ard tillage [START_REF] Macphail | Soil micromorphological evidence of early agriculture in north-west Europe[END_REF]), have to be carefully differentiated from the historical dusty features, also present in the control forest profile (Gebhardt 1990). For example, by examining the relationship and hierarchy of coatings and associated porosity patterns. ln fact today' s European landscape is the result of intensive human activity during prehistoric and historic periods. Modern soils used for experiments are unfortunately often already the product of at least 5,000 years of increasingly intensive cultivation.

The time scale

It is difficult to accelerate experiments. ln the case of Grignon, intensive watering over a short time transformed the natural soil evolution. At the planning stage of the experiment, because the soil water absorbtion capacity was not taken into account, the experimental plots were accidentally transformed into a 'paddy field' soils with atypical spongy microstructure. Even so, this result has been exploited to interpret wet condition in some archaeological soils in Brittany (Gebhardt 1990).

The 'zero time' control sample

Until now (exept for Grignon, see above), micromorphological analysis of experimental cultivated profiles started after at least one year of experimental farming. So we never had the 'zero time' sample, which refers to the soil before experimentation. The only way to obtain a control reference soil was to sample just beside the experimental field. This should provide a good comparison provided that the experiment started less than a couple of years before, and is situated in a relatively 'natural' area (for example a forest in the case of Hambacher Forst, or Lann-Gouh). In Deherain or Butser, experiments are quite mature, and the only 'zero point' available was recently uncultivated grassland along the footpath next to the sampled plots. In all cases, it is diffieult to say how similar the control sample is to a real 'zero time'. It is also quite impossible to experiment in ancient agriculture under the same original conditions found by the first farmers. This does not necessarily matter, however (see Macphail et ul. I990), as the presence of dusty clay in the control forest profile at Hambaeher Forst illustrates that the use of control samples helps to differentiate micromorphological features belonging to the on-site soil history from those caused by experimentation and/or past tillage.

Ageing of microstructure A further question is 'Does microstructure evolve with time?' When the fields are not buried quickly after abandonment, normal pedogenesis may change structure and disturb pedofeatures At Malguénae, for example, after fifty years of abandonment as pasture, ancient tillage features are still recognisable in terms of deep horizon units ploughed up to the surface. On the other hand, typical tillage microporosity has been disturbed by recent bioturbation.

In the case of burial, there may well be structural transformation due to the weight of the overburden (barrow, burial mound, wall, bank). Even now these fundamental influences on palaeosols are not yet fully studied (Macphail and Goldberg this volume). Under the Wareham Experimental Earthwork (Dorset) for example, compaction of the very top of a buried humo-ferric podzol was by thirty per cent (Fisher and Macphail I985), and at Overton Down Experimental Earthwork (Jewell I963;[START_REF] Jewell | The experimental earthwork on Overton Down, Wiltshire, England; the first four years[END_REF]) compaction seems to have occurred rapidly in the last sixteen years of the thirty~two year old experiment (Macphail and Goldberg this volume). In Brittany, prehistoric buried soils were often eroded (Gebhardt I990,I993). Moreover, before burial the organic horizon seems not have been very thick, perhaps having been probably disturbed by deforestation or tillage. This can also make it difficult to compare fresh (unburied) experimental tillage data with cultivation features from archaeological (buried) sites.

Sample representation

The last question is the sampling and environmental factor. In the case of the pig pasture, for example, one sampled area on the slope suggested a strongly colluvial effect. But when we looked at the field, many roots where dug out by the pigs. What kind of profile microstructure would we have observed in an enclosure on flat land'? More sampling at differents levels of the slope wouldprovide a more representative idea of damage to the soil profile caused by pig trampling.

Conclusion

We can conclude by a description of the ideal conditions for experimental research in agrarian archaeology. Experiments should be run in the same area, within a unified pedoclimatic context, similar to the one investigated archaeologically. The ideal experiment should be followed from its beginning with a 'zero time' soil sample and long enough so that the soil has time to record the changes due to the type of cultivation carried out. An example of this is difficult to find because, up until now, experiments in archaeology have often been short and relatively unconcemed with soil changes. Instead of attempting to accelerate experiments, recent and traditional cultivated fields and related features can be studied. These have the advantage of still being well understood by the extant farming population and of having functioned over a sufficiently long period for typical features to develop clearly. Ideally, this should be done in parallel with an historical investigation of farmers who previously farmed the land. Several sampling locations for the same experiment are necessary to obviate sampling uncertainties. In fact, in order to be most efficient, this experimental and ethno-archaeological research should be part of a global regional programme to study the evolution of the arable landscape. Experiments help to answer archaeological questions. Soil processes are very long and studying traditional or extant structures can provide a good opportunity for obtaining answers more quickly. As the historian M. [START_REF] Bloch | Les caractères originaux de l'histoire rurale française[END_REF] said, in order to interpret the past, it is good to look at a past very near the present.

-

  the Dehérain plots (France INA-PG) : established by agronomists on a cambisol (F.A.O. 1975; sol brim: Duchaufour 1983) developed on calcareous colluvium under grassland (INA-PG, unpublished) to study the influence of different types of manuring on crop yield. Several plots have been worked with a cultivator (20 cm deep) or a spade (Ap horizon: 1O cm).

Fig. 1 .

 1 Fig.1. Site location. Archaeological experiments: 1. Butser; 2. Hambacher Forst; 3. Lann-Gouh (Melrand); 4. Grignon Forest. Agronomical experiments: 5. Dehérain plots. Traditional reference sites: 6. St Price (Meirand) and Malguénac.

Fig. 2 .

 2 Fig. 2. The traditional ploughed field (Malguénac, Morbihan) a. (left) Fragments of sharply cut compacted and pale B horizon ploughed up by tillage and are now included in the more dark upper/4p horizon. b.(right) Dusty clay coatings infilling voids (D), observed in the B horizon (25/30cm depth), and related to the ancient cultivation of the field.

Fig. 3 .

 3 Fig. 3. The pig pasture(Lann-Gouh, Morbihan). The compacted colluviated sediment (C) burying the distric cambisol profile of the pig pasture (3b), contrasting with the organic will strongly bioturbated horilons (A0, A1) of the control profile(311). The result of pig activity on the slope was to remove the topsoil and generate a colluvium covering the original profile downslope.

Fig. 4 .

 4 Fig. 4. Comparison of archaeological, traditional and experimental house-floors. (Lann-Gouh). (a,b,c from left to right) The slightly layered surface (4a) has to be related to the traditional earth floor construction. It was constructed by successive spreading of the fine and coarse material on a truncated soil (ie no vegetation and upper horizon ). Root and associated porosity are easily recognisable in the experimental organic house -floor (4b). It results from a complete soil profile (ie with vegetation and upper horizons) which was compacted by daily human and animal trampling. A non-porous medieval house mineral soilfloor (4c), simply compacted by domestic trampling oftlze natural soil after removal of its organic upper horizons.

Acknowledgements

The author wishes to thank all the archaeologists and experimental research workers who allowed access to their material and data. Many thanks to R. I. Macphail for collaborative discussions and help with writing this article and with my English.


Ivailo Hartarsky
email: ivailo.hartarsky@ens.fr

Keywords: Mathematics Subject Classifications: 05D99, 94B65, 60C05

. Furthermore, we discover a link between this problem and a generalisation of the well-known Snake-in-the-Box problem.

Maximal Bootstrap Percolation Time on the Hypercube via Generalised Snake-in-the-Box 1 Introduction

Bootstrap percolation was introduced in 1979 by Chalupa, Leath and Reich [START_REF] Chalupa | Bootstrap percolation on a Bethe lattice[END_REF] as a simplified monotone version of ferromagnetic dynamics and it is in particular related to Glauber dynamics of the Ising model. The general r-neighbour model on a graph G is defined as follows. Consider an initial subset of the vertices (sites) that are declared infected. At each time step every site becomes infected if it has at least r neighbours already infected and infected site always remain such. We say that percolation occurs if eventually all sites of G are infected. In the most classical setting the initially infected sites are selected randomly and independently with probability p and the graph G is taken to be a finite d-dimensional grid {1, . . . , n} d .

One of the founding results in the field was by Aizenman and Liebowitz [START_REF] Aizenman | Metastability effects in bootstrap percolation[END_REF], who determined the order of the critical probability of percolation for r = 2, all fixed d as n → ∞. The simplest setting, r = d = 2 was then studied by Holroyd [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF], who proved that the threshold is sharp and determined the leading term of the critical probability. Further work on that threshold was done and the order of the second term is now known [START_REF] Gravner | Slow convergence in bootstrap percolation[END_REF][START_REF] Hartarsky | The second term for two-neighbour bootstrap percolation in two dimensions[END_REF]. However, the case r > 2 required a lot more care, because the stable sets of infected sites are no longer simple boxes. An important step was done by Cerf and Cirillo [START_REF] Cerf | Finite size scaling in three-dimensional bootstrap percolation[END_REF] and Cerf and Manzo [START_REF] Cerf | The threshold regime of finite volume bootstrap percolation[END_REF], who proved the counterpart of the result of [START_REF] Aizenman | Metastability effects in bootstrap percolation[END_REF]. Their methods were later used in conjunction with Holroyd's to determine the leading term of the critical probability for all fixed r and d when n → ∞ [START_REF] Balogh | The sharp threshold for bootstrap percolation in all dimensions[END_REF].

A less standard and more combinatorial facet of bootstrap percolation consists in keeping n fixed and letting d grow to infinity, so that the simplest case is the high dimensional hypercube. This setting was explored by Balogh and Bollobás [START_REF] Balogh | Bootstrap percolation on the hypercube[END_REF], and later Balogh, Bollobás, and Morris [START_REF] Balogh | Bootstrap percolation in high dimensions[END_REF] determined the critical probability of percolation with high precision for r = 2, and also for high dimensional grids with size not necessarily equal to 2. However, the situation for r > 2 remains entirely open due to the lack of tools to handle the more complicated stable sets, since the method of [START_REF] Cerf | Finite size scaling in three-dimensional bootstrap percolation[END_REF][START_REF] Cerf | The threshold regime of finite volume bootstrap percolation[END_REF] is no longer of relevance.

Alongside the probabilistic perspective on bootstrap percolation, purely combinatorial extremal questions have been widely investigated. Such deterministic bounds have proved useful for obtaining probabilistic results as well, e.g. in [START_REF] Balogh | Bootstrap percolation in high dimensions[END_REF]. However, it has become customary to expect the unexpected, as, more often than not, answers to such extremal questions are very counterintuitive and very far from "common" behaviour. Some deal with the classical 2 dimensional 2-neighbour model, like [START_REF] Benevides | On slowly percolating sets of minimal size in bootstrap percolation[END_REF][START_REF]Maximum percolation time in two-dimensional bootstrap percolation[END_REF][START_REF] Morris | Minimal percolating sets in bootstrap percolation[END_REF], but others [START_REF] Morrison | Extremal bounds for bootstrap percolation in the hypercube[END_REF][START_REF] Riedl | Largest minimal percolating sets in hypercubes under 2-bootstrap percolation[END_REF][START_REF] Morrison | Saturation in the hypercube and bootstrap percolation[END_REF] focus on the hypercube. The typical quantities assessed are the extremal sizes of (extremal) (non-)percolating sets, extremal percolation time or mixtures of those.

One such result by Przykucki [START_REF] Przykucki | Maximal percolation time in hypercubes under 2-bootstrap percolation[END_REF] concerns the maximal percolation time on the hypercube under the 2-neighbour model. Contrary to the result of [START_REF]Maximum percolation time in two-dimensional bootstrap percolation[END_REF] that the maximal percolation time for the same model in two dimensions is of the order of the size of the whole grid considered, for the hypercube of dimension d the maximal percolation time was determined to be merely d 2 3 . Based on the construction in [START_REF] Przykucki | Maximal percolation time in hypercubes under 2-bootstrap percolation[END_REF] one might expect that setting r = 3 would simply allow one to gain another factor of order d and the percolation time to be at most cubic in the dimension. Most surprisingly, we prove that there is a drastic jump between r = 2 and r = 3 for this question. We show that the maximal percolation time goes from close to the trivial lower bound 1, as found in [START_REF] Przykucki | Maximal percolation time in hypercubes under 2-bootstrap percolation[END_REF], to close to the trivial upper bound 2 d . More precisely, we prove that for all r > 2 the maximal percolation time is equal to 2 d d up to a logarithmic factor. The lower bound is based on an entirely explicit construction, though quite elaborate, as it also uses a previously known non-trivial one. An essential ingredient for this bound is a new link we establish between bootstrap percolation and the very well-known snakein-the-box problem, which concerns long induced paths and cycles in the hypercube. It was introduced by Kautz in the late 50s [START_REF] Kautz | Unit-distance error-checking codes[END_REF] and has a wide range of applications, namely in coding, error-correction and others. It was first proved in [START_REF] Evdokimov | The maximal length of a chain in the unit n-dimensional cube[END_REF][START_REF] Glagolev | The minimal coloring of a certain infinite graph[END_REF] that the maximal length of a snake-in-the-box is 2 d up to a constant factor, though its correct asymptotic value is not yet known.

We will rather be concerned with a natural generalisation of the problem, introduced the electronic journal of combinatorics 25(3) (2018), #P3.10 by Singleton in [START_REF] Singleton | Generalized snake-in-the-box codes[END_REF]. It asks for a long path (or cycle) in the hypercube such that sites at distance at least k along the path are also at distance at least k in the hypercube as well. Hence, the snake-in-the-box problem corresponds to k = 2. These paths or cycles are usually referred to as snakes or circuit codes of spread k, but we will call them k-snakes and we will only need 3-snakes for our result. The maximal length of k-snakes was also studied extensively over the last half a century. A very easy upper bound for k = 3, mentioned already in [START_REF] Singleton | Generalized snake-in-the-box codes[END_REF] is

2 d d-2
, is fairly close to the right asymptotics. The right exponent 2 d-o(d) was determined in [START_REF] Kurljandčik | The logarithmic asymptotic behavior of the length of the maximal cycle of the scattering of r > 2[END_REF] and Evdokimov determined the maximal length of a 3-snake to be 2 d d up to a logarithmic factor [START_REF]Chain codes with arbitrary distance[END_REF]. This result is at the base of our construction. For a more recent overview, which is very complete from the mathematical perspective, on snake-in-the-box and related problems, the reader is referred to the survey [START_REF]Circuit codes and the snake-in-the-box problem[END_REF] by the same author.

We should also note that, curiously, another link between bootstrap percolation and the snake-in-the-box (with spread k = 2) problem has been observed in [START_REF] Shende | Maximal induced paths and minimal percolating sets in hypercubes[END_REF], although it is along an entirely different direction and very specific to r = 2.

Notation

In this section, we introduce the notation necessary for the proof of the main result.

We denote by M r (d) our quantity of interest -the maximal time of r-neighbour bootstrap percolation on the d-dimensional hypercube {0, 1} d with its usual graph structure.

Denote by d(•, •) the associated graph distance induced by the norm

x = d i=1 x i for x = (x i ) ∈ {0, 1} d .

Snakes

Definition 1. For k 1 a k-snake is a path (S t ) T t=0 in the hypercube such that, for all t 0 and t ∈ [t + k, T ] it holds that d(S t , S t ) k. We call T the length of S and refer to the parameter t as the time.

Remark 2. For a k-snake of length greater than k this definition implies that for t, t ∈ [0, T ] such that |t -t | k one has d(S t , S t ) = |t -t |. Indeed, each step increases the distance by at most 1, but after k steps we are required to be at distance at least k, so that increasing by 1 was always necessary. Hence, snakes are k-locally isometric to paths. Definition 3. We denote by s(d) the maximal length of a 3-snake in the d-dimensional hypercube.

The following bound was established by Evdokimov [START_REF]Chain codes with arbitrary distance[END_REF]. Hypercubes For the remainder of this paper, we employ the convenient notation used by Przykucki in [START_REF] Przykucki | Maximal percolation time in hypercubes under 2-bootstrap percolation[END_REF]. Though it may appear very technical at first, it will prove itself to be very practical.

Definition 5. For any finite sequence (a

i ) ∈ {0, 1, * } n we denote [a 1 , . . . , a n ] := {(b 1 , . . . , b n ) ∈ {0, 1} n | ∀ 1 i n, a i = * ⇒ b i = a i }
and call all such sets subcubes (of the hypercube {0, 1} n ). We extend this notation to the concatenation of two sequences (a i ) and (b i ) as

[a 1 , . . . , a n ][b 1 , . . . , b k ] := [a 1 , . . . , a n , b 1 , . . . , b k ] .
[a 1 , . . . , a n ] k stands for [a 1 , . . . , a n ] . . . [a 1 , . . . , a n ] repeating k times. We will abusively identify singletons with their unique element, when they arise in this notation, i.e. when * is never used. [0] 3 is its d -6 dimensional subcube whose first three coordinates are 1, 0, and 1 in that order, and whose last three coordinates are all 0.

We may write [1, 0, 1][0] 2 for both the site (1, 0, 1, 0, 0) and the subcube {(1, 0, 1, 0, 0)}.

Definition 7. For any sequence of sequences (a j i )

l j i=1 n j=1
on the alphabet {0, 1, * }, we define their permutation

[a 1 1 , . . . , a 1 l 1 ] . . . [a n 1 , . . . , a n ln ] = σ∈Sn a σ(1) 1 . . . , a σ (1)
l σ(1) . . . a

σ(n) 1 . . . , a σ(n) l σ(n)
where S n is the symmetric group. (0, 0, 0, 1, 0, 0), (0, 0, 0, 1, 0, 1), (0, 0, 1, 0, 0, 0), (0, 0, 1, 0, 0, 1), (0, 1, 0, 0, 0, 0), (0, 1, 0, 0, 0, 1).

It is important to note that the permutation does not act inside each component of the concatenation and moves the whole blocks without interlacing them, so that [ [START_REF] Aizenman | Metastability effects in bootstrap percolation[END_REF]0] 2 are all different sets with 6, 2 and 1 elements respectively.

1] 2 [0] 2 , [1, 1][0, 0] and

The Main Result

In this section, we prove our main result determining the maximal percolation time for all r > 2 in the hypercube up to a log d factor.

Theorem 9. For all r 3 1) .

M r (d) = 2 d d (log d) -O(
the electronic journal of combinatorics 25(3) (2018), #P3.10

The upper and lower bounds are established independently. We start with the lower one, which will follow by linking the bootstrap process to long 3-snakes.

Lemma 10. Let d 15 be odd. Then,

s(d -10) M 3 (d)
Let us sketch the idea before we turn to the proof of the lemma. We would like have a long 3-snake becoming infected one site at a time. To achieve that we fix a long 3-snake in a subcube of codimension 9 and infect neighbours of that snake in new directions in order to have two for each site of the snake. Then we only need to have the beginning of the snake initially infected. We also make sure that next to the end of the snake there is a configuration of lots of infected sites which can percolate only using the end of the snake in addition. Of course, some care is needed in order not to infect any other site by accident before the snake can reach its end.

Proof of Lemma 10. We will need the following technical lemma.

Lemma 11. Let d

6. There is a 3-snake S of dimension d and length T such that the following conditions all hold.

1.

S T -3 = [1, 0, 1, 0, 1][0] d -5 . 2. S T -2 = [1, 0, 1][0] d -3 . 3. S T -1 = [1][0] d -1 . 4. S T = [0] d .
5. S t > 3 for every t < T -3.

T s(d -1).

Proof of Lemma 11. It suffices to satisfy conditions 4-6 and then to permute the coordinates to also fulfil the other conditions. One can achieve conditions 4-6 as follows.

Let S be a d -1-dimensional 3-snake of maximal length with S s(d -1) = [0] d -1 (to obtain it compose a d -1-dimensional 3-snake of maximal length by a suitable isomorphism of the hypercube). Then we set S t = [1]S t for all 0 t s(d -1), T = s(d -1) + 1 and S T = [0] d . Conditions 4 and 6 are clearly satisfied. Furthermore, since S is a 3-snake ending in [0] d -1 , we have that S t 3 for all t s(d -1) -3. Thus, condition 5 does hold by construction and S is indeed a 3-snake.

For convenience denote d := d-3 and d := d-9. Let S be as provided by Lemma 11. Let the initial set of infected sites I be defined as follows (see Figure 1).

• Infect [0] 9 S 0 .

the electronic journal of combinatorics 25(3) (2018), #P3.10 • For i ∈ {1, 2, 3} set

I 0 J 3 J 2 ∅ J 3 J 1 J 1 J 2
S i := S T -i-3j , 0 j T -i 3 . Infect I 0 := [0] 3 [0][1][0] 4 S 1 ∪ [0] 5 [0][1][0] 2 S 2 ∪ [0] 7 [0][1]S 3 .
Do note that we do not include neighbours of the end of the snake [0] 9 S T .

• Moreover, infect

J 1 := [1, 1][ * ] d +1 , J 2 := [0][1][1][0] d and J 3 := [0][1][0][1, 1][0, 0] d -2 2 . Recall that [1, 1][0, 0] k = 0 l k [0, 0] l [1, 1][0, 0] k-l .
We claim that [0] 9 S is infected one site at a time, that no site outside [0] 9 S is infected strictly before S T and that percolation occurs. However, before we turn to the proof of those claims, let us establish some properties of the configuration.

• I 0 , J 2 and J 3 have pairwise no common neighbours. Indeed, -J 2 and J 3 have no common neighbours by parity.

-J 2 and I 0 have no common neighbours, since [0] d ∈ I 0 and two of the first three coordinates are different.

-J 3 and I 0 have no common neighbours. To see this, consider a site j in J 3 at distance 2 from i ∈ I 0 . Those two differ in one of the first three coordinates, so

i has a neighbour in [0] 3 [1, 1][0, 0] d -2
2 . Then i ∈ {1, 3} and by condition 5 of Lemma 11 and parity i is necessarily a neighbour of [0] 9 S T -2 (recall that I 0 does the electronic journal of combinatorics 25(3) (2018), #P3.10

not contain neighbours of [0] 9 S T ). Hence, i ∈ [0] 3 [0] 5 [1][1, 0, 1][0] d -3 . Notice that i necessarily has 2 adjacent 1s, since it has a neighbour in [0] 3 [1, 1][0, 0] d -2 2 .
However, this is the case only

if i = [0] 8 [1, 1, 0, 1][0] d -3 , which has no neighbour in [0] 3 [1, 1][0, 0] d -2 2
-a contradiction.

• The only couples of sites in J 3 at distance (at most) 2 are of the form ([1, 0, 0]x,

[0, 1, 0]x) for some x ∈ [1, 1][0, 0] d -2
2 . Indeed, if the first two coordinates differ, the distance is at most 2 only if all other coordinates are identical and if they do not differ, sites in J 3 are at distance 4.

• Every site i 1 ∈ I 0 has a unique other site i 2 ∈ I 0 at distance (at most) 2. Indeed, consider d(i 1 , i 2 ) 2 and argue that i 1 and i 2 only differ in coordinates 4-9. If the last d coordinates differ, by less than 3, as S is a 3-snake, the time in the snake has different remainder modulo 3 for the two sites and thus, 2 of the the first 9 coordinates must differ. If the last d coordinates differ by 3 or more, S being a 3-snake implies d(i 1 , i 2 ) 3. Clearly, there is a unique site which differs from i 1 only in coordinates 4-9.

• [0] 9 S t has common neighbours with J 3 only for t = T -1. Indeed, by condition 5 of Lemma 11 and parity one has t ∈ {T -1, T -3}. For T -3 it suffices to note that [0] 9 S T -3 has no two consecutive 1s.

• [0] 9 S t has common neighbours with J 2 only for t = T (More generally, [0] 3 a is at distance 2 from J 2 only for a = [0] d ).

Claim 1 At time 0 t < T the set of infected sites is I ∪ {[0] 9 S t , t t}.

Proof of Claim 1. We proceed by induction.

Base: We show that I \ {[0] 9 S 0 } is stable i.e. no uninfected site has three infected neighbours. Consider an uninfected site s and split the reasoning in cases depending on s.

• If s ∈ [0, 0, 1][ * ] d , then it has at most two neighbours in J 2 (since |J 2 | = 2) and at most one neighbour in I 0 . However, J 2 and I 0 have no common neighbours, so it has at most 2 infected neighbours.

• If s ∈ [0][1][1][ * ] d \ J 2
, then it has one neighbour in J 1 (since this is a subcube), at most one neighbour in J 2 , at most one neighbour in J 3 and no neighbours in I 0 . However, J 2 and J 3 have no common neighbours, so it has at most 2 infected neighbours.

• If s ∈ [0][1][0][ * ] d \ J 3
, then it has one neighbour in J 1 (since this is a subcube), at most one neighbour in each of I 0 , J 2 and J 3 . Indeed, for J 3 we know that all sites with (at least) two neighbours in J 3 are not in [0] [START_REF] Aizenman | Metastability effects in bootstrap percolation[END_REF][0][ * ] d . However, I 0 , J 2 and J 3 have pairwise no common neighbours, so we are done.

the electronic journal of combinatorics 25(3) (2018), #P3.10

• If s ∈ [0] 3 [ * ] d \ I 0 ,
then it has no neighbours in J 1 or J 2 and at most two in J 3 , but since J 3 and I 0 have no common neighbours, it suffices to prove that s cannot have 3 neighbours in I 0 . However, we know that each site in I 0 has common neighbours with only one other site in I 0 , which concludes the proof of the base.

Step: Assume that at time 0 t < T -1 the infected sites are I ∪ {[0] 9 S t , t t}. We only need to check that none of the uninfected neighbours of [0] 9 S t other than [0] 9 S t+1 has 3 infected neighbours at time t.

As we know, J 2 and J 3 have no common neighbours with [0] 9 S t , so they cannot contribute. In [0] 9 [ * ] d the only infected site with neighbours in common with [0] 9 S t is [0] 9 S t-2 (or none if t 1), as S is a 3-snake. But their common neighbour different from [0] 9 S t-1 has no other infected neighbours in [0] 9 [ * ] d (since S is a 3-snake), does not neighbour J 1 (since it is in [0] 9

9 S t is [1, 1][0] 7 S t ∈ J 1 .
Those common neighbours being outside J 1 and [0] 3 [ * ] d , they cannot have more than 2 infected neighbours in those two subcubes, which exhausts all possible cases and completes the induction step.

Claim 2 The set J 1 ∪ J 2 ∪ J 3 ∪ {[0] 9 S T -1 } percolates.
Proof of Claim 2. We have the following infections (we do not claim that they happen at different times or in this order).

• [0][1][0] 7 [1][0] d -1 is infected by [0] 9 S T -1 , J 1 and J 3 . • [0][1][0] d +1
is infected by the previous one, J 1 and J 2 .

• [0][1][0][1][0] d -1 is
infected by the previous one, J 1 and J 3 .

• [0][1][0][ * ] d is
infected by the previous one and J 1 . Indeed, for all 2 k d every site in

[0][1][0][1] k [0] d -k has at least 2 neighbours in [0][1][0][1] k-1 [0] d -k+1
and one neighbour in J 1 , so those sets become infected successively by induction.

• [0] 3 [ * ] d is

Conclusion and open problems

In conclusion, our result exhibits a significant difference between the 2 and 3-neighbour models on the hypercube. The reason why our method does not work for the 2-neighbour case is that parasite infections are inevitable. More precisely, there necessarily appear additional infections around an infected path -at each 'corner' of the path (in the hypercube a path has 'corners' at each step) at the first time step and more afterwards. Further understanding of the different behaviours should be of use in attacking the 3-neighbour model on the hypercube with random initial condition, by showing what anomalies one needs to take into consideration. We list here a few of the questions raised by the present work, not necessarily aiming directly at solving that model.

The first natural question to ask in view of our work is to determine the exact order of the maximal percolation time. We conjecture that the upper bound is tight up to a constant. It should be noted, that this result would follow from the same proof, if one establishes the corresponding lower bound for the maximal length of 3-snakes, improving the result of [START_REF]Chain codes with arbitrary distance[END_REF].

Secondly, a probably difficult question is to determine the random percolation time. The probabilistic counterpart of our extremal result would be as follows.

Question 15. Conditionally on percolating, what is the order of the percolation time if the initially infected sites are chosen randomly and independently with probability p(d)?

It would, namely, be interesting to see if exponentially large times such as the ones we give manage to alter the mean percolation time despite their low probability of occurrence.

Finally, in view of the more recent development of U-bootstrap percolation in 2 dimensions [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF][START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF] and, currently in higher, but fixed number of dimensions, one could ask for similar results about models more general than the r-neighbour model, but still on the hypercube (e.g. a site is infected if some fixed subset, defined up to isomorphism, of its 2neighbourhood is infected). An answer of satisfactory generality to the following question might need to wait until U-bootstrap percolation setting is extended to the hypercube, but it is worth investigating nonetheless.

Question 16. When is the order of the maximal time of U-bootstrap percolation on the hypercube up to a constant given by the maximal length of a k-snake for some k and how is k determined by U? the electronic journal of combinatorics 25(3) (2018), #P3.10

Proposition 4 .

 4 For all d 3 s(d) 2 d d(log d) 2 . the electronic journal of combinatorics 25(3) (2018), #P3.10

Example 6 .

 6 The hypercube of dimension d is thus denoted by [ * ] d and [1, 0, 1][ * ] d-6

Example 8 .

 8 The elements of [0][0] 2 [1, 0][ * ] are

Figure 1 :

 1 Figure 1: Each vertex of the cube in this picture represents a d dimensional hypercube, so that only the first three dimensions of [ * ] d are visible. We indicate the positions of the different parts of the initial infected set I. J 1 consists of two entire d -dimensional subcubes, J 2 has one site in each of the two subcubes indicated, J 3 has d 2 sites at distance 4 in each of the two subcubes indicated. Finally, I 0 contains two neighbours of each site in the 3-snake [0] 9 S (except its end). The 3-snake in question lies in the same d dimensional subcube as I 0 .

  [ * ] d ), and nor does it neighbour I 0 (since the only neighbours of I 0 in [0] 9 [ * ] d are in [0] 9 S by construction). Furthermore, the only other infected sites in [0] 3 [ * ] d \[0] 9 [ * ] d with common neighbours with [0] 9 S t are the 4 neighbours of [0] 9 S t±1 in I 0 . Recall that each of those has a common neighbour only with one other, so the only sites with three neighbours among those four and [0] 9 S t are [0] 9 S t±1 . Moreover, J 1 does not contribute, as before, because those 4 sites are in [0] 3 [ * ] d . Finally, the only infected site outside [0] 3 [ * ] d with common neighbours with [0]

  infected by the previous one and [0] 9 S T -1 . Indeed, all sites in [0]3 [ * ] d have two infected neighbours from the previous step, so they only need one more in order to be infected. But since [0] 3 [ * ] d is connected and contains the infected site [0] 9 S T -1 , it does become infected entirely.• [0][1][1][ * ] d is infected by [0][1][0][ * ] d , J1 and J 2 just like in the previous step. • [0, 0][1][ * ] d is infected by the ones in the previous two steps. the electronic journal of combinatorics 25(3) (2018), #P3.10

Conjecture 14 .

 14 Prove that for all fixed r 3M r (d) = Θ 2 d d .

Acknowledgements

The author would like to thank Stephen G. Z. Smith -for numerous particularly helpful discussions and encouragement, Dmitry Chelkak -for crucial bibliographic help and Lyuben Lichev -for meticulous proofreading.

Hence, the whole hypercube is infected.

The lemma follows trivially from the two claims.

The next lemma establishes our upper bound on the percolation time. Proof. Assume that M r (d) > (4r + 2) 2 d d for some d and consider a percolating set of initially infected sites, which achieves the maximal time. For each site v of the hypercube denote t v its percolation time. Note that any site v has at most r -1 neighbours u such that t v -t u > 1, so there are at most (r -1)2 d edges uv of the hypercube such that

and good otherwise. Thus, there are at most 2 d+2 (r-1) d bad sites in total, since an edge contributes to at most 2 of them. But then there are at most 2 d+2 (r-1) d values of t when a bad site becomes infected and in particular there are more than

values when a good site becomes infected. But if v is a good site, then at least d 2 of its neighbours are infected at time t v -1, t v or t v + 1. Hence, applying this to one good site for each time when there is one, one obtains that there are more than

infected sites, since each one is counted up to three times -a contradiction.

Remark 13. In order to obtain a better constant with the same proof, bad sites should be defined to have Cd edges of the type specified and C should then be optimised.

The main result now follows immediately.

Proof of Theorem 9. Let us first prove the lower bound. For r > 3, consider a configuration giving M 3 (d -r + 3) in a (d -r + 3)-dimensional subcube and infect the rest of the hypercube. Then sites in that subcube follow exactly the 3-neighbour bootstrap process restricted to it and thus the problem is reduced to r = 3. For r = 3 the result follows directly from Lemma 10 and Proposition 4, so we are done when d is odd. Consider


Marcu-Antone Orsoni

Reachable states and holomorphic function spaces for the 1-D heat equation

published or not. The documents may come

Reachable states and holomorphic function spaces for the 1-D heat equation

Introduction

Problem setting

Denote by X " W ´1,2 p0, πq, the dual of the Sobolev space W 1,2 0 p0, πq. We consider the following boundary control problem of the heat equation. For any u :" pu 0 , u π q P L 2 loc pp0, `8q, C 2 q -the so-called input (or control) function -and f P X, this equation admits a unique solution y P C pp0, `8q, Xq (see [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]Prop. 10.7.3]) defined by @t ą 0, ypt, ¨q " T t f `Φt u

(1)

where pT t q tě0 is the Dirichlet Laplacian semigroup and Φ t is the controllabililty operator (see [TW09, Prop. 4.2.5]). For f P X and τ ą 0, we will say that g P X is reachable from f in time τ if there exists a boundary control u P L 2 pp0, τ q, C 2 q such that the solution of (HE) satisfies ypτ, ¨q " g. We denote by R f τ the set of all reachable fonctions from f in time τ . Because of the smoothing effect of the heat kernel, it is clear that for arbitrary control u P L 2 pp0, τ q, C 2 q, we cannot reach any non-regular functions. So, R f τ Ĺ X. In other words, the equation (HE) is not exactly controllable for any time τ ą 0. It is thus natural to seek for more precise information on R f τ . First of all, we remind that this set has some invariance properties. Indeed, [START_REF] Yu | Some problems in the theory of optimal control[END_REF] and [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF] have shown that the heat equation (HE) is null-controllable in any time (see [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], [START_REF] Fursikov | Controllability of evolution equations[END_REF] or [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]Prop. 11.5.4] for the n-dimensional case). This means that : @f P X, @τ ą 0, 0 P R f τ . It is clear from (1) that this latter condition is equivalent to Ran T τ Ă Ran Φ τ . So using (1) again we obtain R f τ " Ran Φ τ , which means that R f τ does not depend on the initial condition f P X. Thus, we can take f " 0 and write R τ " R 0 τ . Moreover, since Φ τ P LpL 2 pr0, τ s, C 2 q, Xq, R τ is a linear space, named reachable space of (HE). Finally, the null-controllability in any time τ ą 0 implies also that this space does not depend on time τ ą 0 (see [START_REF] Fattorini | Reachable states in boundary control of the heat equation are independent of time[END_REF], [START_REF] Seidman | Time-invariance of the reachable set for linear control problems[END_REF], or [HKT17, Rmk 1.1]). Note that these two invariance properties hold for every linear control system wich satisfies the null-controllability in any positive time.

Notations

In the rest of the paper, we denote by D " tz P C | |z| ă 1u the unit disc and by C `" tz P C | Imz ą 0u the upper-half plane. Let Ω be a simply connected domain in the complex plane with at least two boundary points. Write HolpΩq for the algebra of holomorphic functions on Ω. We say that f P HolpΩq belongs to the Hardy space H p pΩq p0 ă p ă `8q if the subharmonic function |f | p admits a harmonic majorant on Ω. We say that f P HolpΩq belongs to the Smirnov space E p pΩq p0 ă p ă `8q if there exists a sequence pγ n q nPN of rectifiable Jordan curves eventually surrounding each compact subdomain of Ω such that the space E p pΩq consists of all the functions holomorphic on C `such that

}f } p p " sup yą0 ż R |f px `iyq| p dx ă 8.
This space is often called the Hardy space of the upper-half plane ([Gar07], [START_REF] Nikolski | Operators, Functions, and Systems: An Easy Reading[END_REF], [START_REF] Ya | Lectures on Entire Functions[END_REF], [START_REF] Koosis | Introduction to H p Spaces[END_REF]), and denoted by H p pC `q. Assume now that Ω is a domain bounded by a rectifiable Jordan curve γ. In this case, each f P E p pΩq (1 ď p ă 8) admits a non-tangential limit almost everywhere on γ (denoted again by f ) which belongs to L p pBΩq, and satisfies the Cauchy formula

@z P Ω, f pzq " 1 2iπ ż γ f puq u ´z du.
2

With this in mind, we will say that f P HolpΩq belongs to the Smirnov-Zygmund space E L log `L pΩq if f P E 1 pΩq and its non-tangential limit on γ belongs to L log `LpBΩq, that is ş γ |f pzq| log `p|f pzq|q|dz| ă `8. Therefore, the following inclusions are clear

@1 ă p ă `8, E p pΩq Ă E L log `LpΩq Ă E 1 pΩq.
For more details on the Hardy space and the Smirnov space, we refer to [Dur70, Chap. 9 et 10]. For the cases of the disc and the upper-half plane, see also [START_REF] Rudin | Real and Complex Analysis[END_REF], [START_REF] Garnett | Bounded Analytic Functions[END_REF], [START_REF] Nikolski | Operators, Functions, and Systems: An Easy Reading[END_REF], [START_REF] Ya | Lectures on Entire Functions[END_REF], [START_REF] Koosis | Introduction to H p Spaces[END_REF].

Finally, the weighted Bergman space A p pΩ, ωq, where ω is a non negative mesurable function on Ω, consists of all functions f P HolpΩq such that

}f } p A p pΩ,ωq " ż Ω |f px `iyq| p ωpx `iyqdxdy ă `8.
When ω " 1, A p pΩ, ωq is the classical Bergman space which we simply denote A p pΩq.

Scientific context

It seems that the first work on this problem is in [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF]. Using a moment method, Fattorini and Russel showed that if there exists A ą 0 and B ą 0 such that

@n P N ˚, |a n | ď A expp´pπ `Bqnq (2)
then the function defined by gpxq " ř `8 n"1 a n sinpnxq is reachable. For δ ą 0, denote by H δ the space of continuous functions which are π-periodic on R, which extend holomorphically on the strip |Imz| ă δ and whose derivatives of even orders vanish in 0 and in π. Adjusting a discrete Paley-Wiener theorem[QZ13, Chap.IV, sect.V, Thm V.1 vi), p. 98] to the orthonormal basis psinpnxqq ně1 , we obtain from (2) that for δ large enough, H δ is included in Ran Φ.

Later, Martin, Rosier and Rouchon improved this result in [MRR16, Thm 1.1]. On the one hand, they showed that the holomorphic functions on the disk

B " ! z P C ˇˇ z ´π 2 ă π 2 e p2eq ´1 )
are reachable. On the other hand, they proved that the reachable functions extend holomorphically to the square (see Figure 1)

D " ! z " x `iy P C ˇˇ x ´π 2 `|y| ă π 2
) .

To summarize, we have HolpBq Ă R τ Ă HolpDq. Dardé and Ervedoza improved again this latter result in [DE18, Thm 1.1] showing that all the functions which are holomorphic on a neighborhood of D are reachable. This result combined with the previous result means that Ran Φ τ is a space of holomorphic functions on D. Finally, the best known result on this problem to our knowledge is given in [START_REF] Hartmann | From the reachable space of the heat equation to hilbert spaces of holomorphic functions[END_REF], where the authors proved that the reachable space is sandwiched between two Hilbert spaces of holomorphic functions on the square D. More explicitly, it satisfies the inclusions

E 2 pDq Ĺ Ran Φ τ Ă A 2 pDq (3)
(see the previous subsection for the definitions). Key tools used in that paper include a unitary Laplace type integral operator studied by Aikawa, Hayashi and Saitoh [START_REF] Aikawa | The Bergman space on a sector and the heat equation[END_REF], as well as a Riesz basis of exponentials in E 2 pDq discussed by Levin and Lyubarskii [LL75]. The idea of our paper is to avoid the Riesz basis of exponentials and to use complex analysis tools like Cauchy formula, Hilbert transform and Bmethods which will allow us to improve significantly -in terms of function spaces of complex analysis -the lower bound E 2 pDq.

Main results

Let ∆ " z P C ˇˇ| argpzq| ă π 4 ( . The first central result of this paper is the following explicit characterization of the reachable space.

Theorem 1.1. We have Ran Φ τ " A 2 p∆q `A2 pπ ´∆q.

We mention another characterization which was very recently observed by T. Normand (see [Tuc]). Denote by ω 0 and ω π the weights defined by @z P ∆, ω 0 pzq " e

Repz 2 q 2τ τ and @z P π ´∆, ω π pzq " ω 0 pπ ´zq,

then Ran Φ τ " A 2 p∆, ω 0 q `A2 pπ ´∆, ω π q, (5) independently of τ ą 0. Note that the inclusion "Ą" in (5) was already known in [START_REF] Hartmann | From the reachable space of the heat equation to hilbert spaces of holomorphic functions[END_REF].

To prove the Theorem 1.1, the main idea is to write a certain integral operator as a Laplace type transform and to use a Paley-Wiener type theorem.

The central question raised by the above result is the description of the sum A 2 p∆q `A2 pπ ´∆q. We obviously have

A 2 p∆q `A2 pπ ´∆q Ă A 2 pDq, (6)
and from the results in [START_REF] Hartmann | From the reachable space of the heat equation to hilbert spaces of holomorphic functions[END_REF] it also follows that

E 2 pDq Ă A 2 p∆q `A2 pπ ´∆q,
but how can we decide in general whether a given holomorphic function on D, not necessarily in E 2 pDq, can be written as a sum of two functions in the Bergman spaces on ∆ and π ´∆ respectively? Note that this is a very natural complex analysis question which can now be discussed completely disconnected from the initial control problem. Such a type of problem is related, for instance, to the so-called First Cousin Problem (see [AM04, Thm 9.4.1]), which in our situation turns into a specific Cousin Problem with L 2 -estimates. We will call this problem the First Cousin Problem for Bergman spaces. Following the proof of the classical First Cousin Problem of [START_REF] Amar | Analyse complexe. Cassini[END_REF] and using Hörmander L 2 -estimates for the B-equation, we prove the following partial answer to the description of A 2 p∆q `A2 pπ ´∆q.

Theorem 1.2. Let z 0 " π 2 `i π 2 and z 1 " z 0 " π 2 ´i π 2 be the upper and lower vertices of D. Then A 2 `D, |pz ´z0 qpz ´z1 q| ´2˘Ă A 2 p∆q `A2 pπ ´∆q.

In view of (6) Theorem 1.2 gives an optimal result for functions in the sum of the two Bergman spaces outside z 0 and z 1 . However the constraints in z 0 and z 1 are rather strong, implying in particular that functions in A 2 `D, |pz ´z0 qpz ´z1 q| ´2ȃ re locally bounded at these points.

Our second result, involving completely different tools, allows to show that functions with almost characteristic Bergman space growth, in particular at z 0 and z 1 , are also in the sum.

Theorem 1.3. E L log `LpDq Ă A 2 p∆q `A2 pπ ´∆q.
Note that in view of Theorem 1.1, this result improves the left inclusion (3) obtained in [START_REF] Hartmann | From the reachable space of the heat equation to hilbert spaces of holomorphic functions[END_REF] since E 2 pDq Ă E L log `LpDq. The methods used in the proof of Theorem 1.3 are for the most part harmonic and complex analysis methods. More precisely, we use essentially the Cauchy formula for Smirnov functions, a local regularity result for the Cauchy Transform on the upper-half plane and the embedding H 1 pDq Ă A 2 pDq due to Hardy and Littlewood.

It should be observed that in terms of growth of functions in the different spaces, this result is almost sharp (especially close to the points z 0 and z 1 which are not covered by Theorem 1.2). Indeed, a function in A 2 pDq cannot grow faster than 1 dpz,BDq , while the growth of a function in E L log `LpDq is bounded by 1 dpz,BDq logp 1 dpz,BDq q . This last estimate comes from the Cauchy formula and the Hölder inequality for Orlicz spaces.

As an easy consequence of the above results we would like to state the following corollary.

Corollaire 1.4. We have

E L log `LpDq `A2 `D, |pz ´z0 qpz ´z1 q| ´2˘Ă A 2 p∆q `A2 pπ ´∆q Ă A 2 pDq.
In view of Theorem 1.1, this corollary thus improves (3) found in [START_REF] Hartmann | From the reachable space of the heat equation to hilbert spaces of holomorphic functions[END_REF]. Nevertheless, though A 2 `D, |pz ´z0 qpz ´z1 q| ´2˘functions behave like arbitrary A 2 pDqfunctions outside z 0 , z 1 , and E L log `LpDq-functions allow a growth in a sense close to A 2 pDq-functions in z 0 , z 1 , the above corollary still leaves of course a gap. Note that there is no natural completion of A 2 `D, |pz ´z0 qpz ´z1 q|

´2˘i n A 2 pDq.
The paper is organized as follows. In Section 2, we prove Theorem 1.1. Sections 3 and 4 are devoted to the proofs of Theorems 1.3 and 1.2 respectively.

Sum of Bergman spaces

We start recalling some facts from [START_REF] Hartmann | From the reachable space of the heat equation to hilbert spaces of holomorphic functions[END_REF]. First, it is not difficult to check that the reachable states of the 1-D heat equation can be represented as a Fourier-sine series in the following way:

pΦ τ uqpxq " 2 π ÿ ně1 n "ż τ 0 e n 2 pσ´τ q u 0 pσq dσ  sinpnxq `2 π ÿ ně1 np´1q n`1
"ż τ 0 e n 2 pσ´τ q u π pσq dσ  sinpnxq, τ ą 0, x P p0, πq.

With the elementary formula sin u " pe iu ´e´iu q{p2iq in mind and the Poisson summation formula, the authors of [START_REF] Hartmann | From the reachable space of the heat equation to hilbert spaces of holomorphic functions[END_REF] show that

pΦ τ uqpxq " ż τ 0 BK 0 Bx pτ ´σ, xqu 0 pσq dσ `ż τ 0 BK π Bx pτ ´σ, xqu π pσq dσ,
where

K 0 pσ, xq " ´2 π ˜ÿ ně1 e ´n2 σ cospnxq `1" ´1 π ÿ ně1 e ´n2 σ `einx `e´inx ˘´2 π " ´1 π ÿ nPZ ˚e´n 2 σ e inx ´2 π
, σ ą 0, x P p0, πq.

Hence, setting Ă K 0 pσ, zq " ´b

1 πσ ř mPZ ˚e´p z`2mπq 2 4σ
, we can write (see [HKT17, equation (2.18)])

Φ τ pu 0 , u π q " r Φ τ u 0 `r r Φu π `R0,τ u 0 `Rπ,τ u π (7)
where

" r Φ τ f ı psq " ż τ 0 se ´s2 4pτ ´σq 2 ? πpτ ´σq 3 2 f pσqdσ and " r r Φ τ f  psq " " r Φ τ f ı pπ ´sq rR 0,τ f s psq " ż τ 0 B Ă K 0
Bs pτ ´σ, sqf pσqdσ and rR π,τ f s psq " rR 0,τ f s pπ ´sq.

We can now prove Theorem 1.1.

Proof of Theorem 1.1. The key of the proof is that r Φ τ is an isometry from L 2 p0, τ q to A 2 p∆q and we can compute its range. Indeed, denote by L the normalized Laplace transform defined by Lpf qpsq " 1 ? π ş `8 0 e ´st f ptqdt and G : A 2 pC `q Ñ A 2 p∆q the unitary operator associated to the conformal mapping z Þ Ñ z 2 from ∆ to C `, defined by Gpf qpzq " 2zf `z2 ˘. By the change of variables t " 1 4pτ ´σq , we obtain

@s P ∆, ´r Φ τ f ¯psq " ż τ 0 se ´s2 4pτ ´σq 2 ? πpτ ´σq 3 2 f pσqdσ " s ? π ż `8 1 4τ e ´s2 t ? t f ˆτ ´1 4t ˙dt Define for f P L 2 p0, τ q, pT f qptq " # f pτ ´1 4t q 2 ? t if t ą 1 4τ , 0 if 0 ă t ď 1 4τ .
.

It is easily seen that the operator T is an isometry from L 2 p0, τ q to L 2 pR `, dt t q with range L 2 ``1 4τ , `8˘, dt t ˘. Hence ´r Φ τ f ¯psq " 2sLpT f q `s2 ˘" pGLT f q psq.

The last step is the following Paley-Wiener type theorem for Bergman spaces (which seems to be a "folk theorem", a proof for which may be found in [START_REF] Duren | A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces[END_REF]).

Proposition 2.1. The Laplace transform L is unitary from L 2 `R`,

dt t ˘to A 2 pC `q where C `" tz | Rez ą 0u.
So, if » means that the operator is unitary, we have the following diagram.

L 2 p0, τ q T ÝÑ » L 2 ˆˆ1 4τ , `8˙, dt t ˙Ă L 2 ˆR`, dt t ˙L ÝÑ » A 2 pC `q G ÝÑ » A 2 p∆q
Hence, by composition, r Φ τ is isometric from L 2 p0, τ q to A 2 p∆q, and

Ran r Φ τ " GL " L 2 ˆˆ1 4τ , `8˙, dt t ˙ Ă A 2 p∆q.
We get a similar result for r r Φ τ . In order to discuss the range of Φ τ we thus have to investigate the remainder terms R 0,τ and R π,τ , which, morally speaking, are sums converging very quickly since they involve gaussians centered essentially at πn, n P Z ˚. For these remainder terms we will use the lemma below which is a straightforward modification of Lemma 4.1 of [START_REF] Hartmann | From the reachable space of the heat equation to hilbert spaces of holomorphic functions[END_REF], the main difference being a square root in the integral operator, which does not change the boundedness and the convergence to zero. Lemma 2.2. Let ω 0 and ω π be the weights defined in (4). Then R 0,τ and R π,τ are bounded from L 2 p0, τ q to A 2 p∆, ω 0 q`A 2 pπ´∆, ω π q. Moreover, }R 0,τ } " }R π,τ } Ñ τ Ñ0 0.

Since A 2 p∆, ω 0 q `A2 pπ ´∆, ω π q Ă A 2 p∆q `A2 pπ ´∆q, the inclusion Ran Φ τ Ă A 2 p∆q `A2 pπ ´∆q is a direct consequence of the decomposition (7), the above discussion and Lemma 2.2.

For the converse inclusion, we will prove A 2 p∆q Ă Ran Φ τ and A 2 pπ ´∆q Ă Ran Φ τ . Using that G and L are unitary, we have the decomposition

A 2 p∆q " GL " L 2 ˆR`, dt t ˙ " GL " L 2 ˆˆ0, 1 4τ
˙, dt t ˙'L 2 ˆˆ1 4τ , `8˙, dt t ˙ " X 0 ' Ran r Φ τ
where we wrote X 0 :" GL " L 2 ``0, 1 4τ ˘, dt t ˘‰ and where, as usual ' means orthogonal sum. Similarly, we have A 2 pπ ´∆q " X π ' Ran r r Φ τ , where X π is the image of X 0 by the transformation f Þ Ñ f pπ ´¨q. Hence, it is enough to prove that X 0 , X π , Ran r Φ τ and Ran r r Φ τ are contained in Ran Φ τ . For this, note that for every u 0 P L 2 p0, τ q, we have r Φ τ u 0 " Φ τ pu 0 , 0q ´R0,τ u 0 .

Since Ap∆, ω 0 q `A2 pπ ´∆, ω π q Ă Ran Φ τ , we get from Lemma 2.2 that R 0,τ u 0 P Ran Φ τ . It follows that Ran r Φ τ Ă Ran Φ τ . The case of Ran r r Φ τ is similar. Finally, for a ą 0, denote by

X 0 Ă GL " L 2 `0, 1 4τ ˘‰ Ă GL " L 2 `´1 4τ , 1 4τ ˘‰ " G rPW a pC `qs.
Thus, X 0 is a space of entire functions and, as such, is contained in the reachable space. The same argument proves also that the reachable space includes X π , and the proof is complete.

Proof of Theorem 1.3

Let P pzq " z `2iπ. It suffices to prove the following assertion.

@f P E L log `L pDq , f P P A 2 p∆q `A2 pπ ´∆q (8)
Indeed, assume that (8) is true and let g P E L log `L pDq. Since P is bounded analytic on s D, P g belongs also to E L log `L pDq. Hence, by (8), g " pP gq{P belongs to A 2 p∆q `A2 pπ ´∆q and then to Ran Φ τ by Theorem 1.1, which proves the inclusion. Remark 3.1. With a more refined argument, as used in [HKT17, corollary 3.6] and here in Section 4, we can prove that E L log `L pDq Ă A 2 p∆, ω 0 q `A2 pπ ´∆, ω π q where ω 0 and ω π are defined in (4). This observation also follows from Thm 1.3 and Normand's result (5) (see [Tuc]) So, pick f P E L log `L pDq and let us prove (8).

Decomposition. Let γ be the boundary of D parameterized counterclockwise side by side as follows (see Figure 1). The key idea is to decompose f via the Cauchy formula for functions in E 1 pDq (see [Dur70, Thm. 10.4 p.170]) :

γ 1,`: r0, 1s ÝÑ C t Þ ÝÑ p1 ´tq π 2 p1 `iq γ 1,´: r0, 1s ÝÑ C t Þ ÝÑ p1 ´iq π 2 t γ 2,`: r0, 1s ÝÑ C t Þ ÝÑ πp1 ´tq `t ´p1 `iq π 2 ¯γ2,´: r0, 1s ÝÑ C t Þ ÝÑ p1 ´iq π 2 p1 ´tq `tπ ą ą ă ă 0 D γ 1,´γ2,´π γ 2,γ 1,`∆
@z P D, f pzq " 1 2iπ ż γ f puq u ´z du " 1 2iπ ÿ kPt1, 2u εPt˘u ż γ k,ε f puq u ´z du " 1 2 ÿ kPt1, 2u εPt˘u f k,ε pzq
where we have written

f k,ε pzq " 1 iπ ż γ k,ε f puq u ´z du, k P t1, 2u , ε P t˘u.
For the reader acquainted with Hardy spaces, the crucial observation here is that f k,ε can be seen -modulo rotation and translation -as a scalar product between a (compactly supported) function and a reproducing kernel of the Hardy space, which thus yields a (Riesz-) projection on the Hardy space. It is known that this projection is bounded when f k,ε P L p , p ą 1, but not when p " 1. As will be explained below, on the real line, this boundedness remains valid when f k,ε P L log L and f k,ε is compactly supported. Once we have established this fact, a theorem by Hardy and Littlewood on inclusion between Hardy and Bergman spaces will allow to conclude.

The remainder part of the section will be devoted to show that f 1,ε {P P A 2 p∆q and f 2,ε {P P A 2 pπ ´∆q for ε P t˘u. We cut each sector ∆ and π ´∆ in two disjoint parts, which will be treated separately. For that, given a fixed a ą 0, denote by D a the homothetic dilation of D with center 0 and obtained by adding length a ą 0 to the sides of D (see Figure 2). We will consider the disjoint union ∆ " D a Y ∆zD a (and similarly for π ´∆). The proof is composed of two steps. ' Step 1 : In this step we prove the following claim:

f 1,ε {P P A 2 p∆zD a q (9)
(the case f 2,ε {P P A 2 ppπ ´∆q z pπ ´Da qq follows in a similar fashion).

To do so, remark that there exists a constant C a ą 0 such that for any z R D a , |z| `1 ď C a dpz, BDq. Using the triangular inequality, we have for any k P t1, 2u and

ε P t˘u, @z R D a , |f k,ε pzq| ď 1 π ż γ k,ε ˇˇˇf puq u ´z ˇˇˇ| du| ď }f } L 1 pBDq 1 πdpz, BDq ď C |z| `1 ,
where we have used that L log `L Ă L 1 on a segment. So, since ´2iπ R ∆zD a , we obtain

ż ∆zDa f 1,ε pzq ppzq 2 dApzq ď C ż ∆zDa dApzq |2iπ `z| 2 p1 `|z|q 2 ă `8.
This proves claim (9).

'

Step 2 : This step is more delicate and uses the Cauchy (or Hilbert) transform and the inclusion H 1 pDq Ă A 2 pDq.

We need to show the following claim f 1,ε {P P A 2 pD a q (10) (and f 2,ε {P P A 2 pπ ´Da q). It is enough to treat the case f 1,`, the others follow in a similar way. For g P L 1 pRq, we denote by Cg its Cauchy Transform defined by pCgq pzq "

1 iπ ż R gptq t ´z dt, z P C `:" tz P C | Impzq ą 0u .
For more details on this operator, we refer to [START_REF] Cima | The Cauchy Transform[END_REF].

We first explain briefly how to translate f 1,`t o Cg for some suitable g. It is essentially rotating and translating the line through γ 1,`t o R. To be more explicite, let α 1,`: z Þ Ñ 1 `?2 π e i 3π 4 z. This is a direct similarity transformation which sends D a to C `and in particular γ 1,`o nto r0, 1s (note that the orientation is preserved, e.g. the endpoint 0 of γ 1,`i s sent to 1). Let f γ 1,`p tq " 1 r0,1s ptqf pγ 1,`p tqq. For all z P D a , we have

f 1,`p zq " 1 iπ ż γ 1,`f puq u ´z du " 1 iπ ż 1 0 f pγ 1,`p tqq γ 1,`p tq ´z γ 1 1,`p tqdt " 1 iπ ż R f γ 1,`p tq p1 ´tq π 2 p1 `iq ´z ´´π 2 p1 `iq ¯dt " 1 iπ ż R f γ 1,`p tq t ´α1,`p zq dt " ´Cf γ 1,`¯p α 1,`p zqq
So, since P does not vanish on D a , we obtain

ż Da f 1,`p zq P pzq 2 dApzq ď C ż Da |f 1,`p zq| 2 dApzq " C ż Da ´Cf γ 1,`¯p α 1,`p zqq 2 dApzq " C 2 ż α 1,`p Daq ´Cf γ 1,`¯p zq 2 dApzq, ( 11
)
where we have used in the last step that α 1,`i s an affine change of variable with constant jacobian. As already written, α 1,`p D a q is a square in the upper-half plane with a segment of the real line as one of its sides. We will next appeal to the following regularity result of the Cauchy transform which is essentially a combination of a result by Calderon-Zygmund [CZ52, Thm 2, p.100] on the boundedness of the Cauchy transform for compactly supported functions from L log L to H 1 , and a result by Hardy-Littlewood on inclusion between H 1 and A 2 on the disk.

Remark 3.3. Note that in this proposition, we do not need to assume any link between the (compact) support of f and the segment I. However, we will apply later on the result for the case when the support of f is included in I (and I " r0, 1s).

We start with a first intermediate result.

Lemma 3.4. Let f P L log `LpRq having compact support. Then the Cauchy Transform Cf satisfies:

sup yą0 ż L ´L |Cf px `iyq|dx ă `8
This result is certainly known to the experts in harmonic analysis. We include its proof for convenience of the reader. It is essentially based on the following theorem by Calderon and Zygmund [CZ52, Thm 2 p.100]. Let fλ pxq " ş |x´y|ą1{λ f pyq{px ´yqdy. Note that lim λÑ0 fλ pxq corresponds to the Hilbert transform of f . Theorem 3.5 (Calderon-Zygmund). If |f |p1 `log `|f |q is integrable over R, then fλ is integrable over every set S of finite measure. Moreover,

ż S | fλ |dx ď A S ż R |f |p1 `log `|f |qdx `BS ,
where A S and B S are constants depending only on S, but neither on f nor on λ.

Proof of Lemma 3.4. For y ą 0, let P y pxq " y πpx 2 `y2 q and Q y pxq "

x πpx 2 `y2 q be the Poisson and the conjugate Poisson kernels (Q 0 corresponds to the kernel of the Hilbert transform). Then we have @z P C `, Cf pzq " Pf pzq `iQf pzq where we have written Pf px `iyq " pP y ˚f qpxq and Qf px `iyq " pQ y ˚f qpxq. So it suffices to show

sup yą0 ż L ´L |Pf px `iyq|dx ă `8 and sup yą0 ż L ´L |Qf px `iyq|dx ă `8.
The first inequality is clear from classical properties of the Poisson kernel (for this it is even enough that f P L 1 pRq, see [Gar07, Thm 3.1]). Consider the second inequality. Recall the following estimate (see for example [Gar07, p. 105]) @y ą 0, @x P R, Q y f px `iyq ´r f y pxq ď CM f pxq where M f is the Hardy-Littlewood Maximal function and r f y is defined by

@x P R, r f y pxq " ż |t´x|ąy Q y px ´tqf ptqdt.
This, together with a classical result on the regularity of M f (see [START_REF] Garnett | Bounded Analytic Functions[END_REF]p. 23]) and Theorem 3.5 above yields the desired result.

We need some more notation. Let L ą 0 such that

I Ă ‰ ´L 2 , L 2 " and supp f Ă ‰ ´L 2 , L 2 "
. Denote by Ω L the square contained in C `with one side being the segment r´L, Ls. Lemma 3.6. Under the conditions of the proposition, we have Cf P E 1 pΩ L q.

Proof of Lemma 3.6. In order to prove Cf P E 1 pΩ L q, pick pω ε q 0ăεăε 0 , ε 0 ă L{2, a sequence of rectifiable Jordan curves given by the sides of the squares contained in Ω L one side of which is ω ε,0 " r´L `ε, L ´εs `iε and ω ε,1 corresponds to the remaining three sides of the square (see Figure 3). Let ω ε " ω ε,0 _ ω ε,1 (concatenation of the two Jordan curves, orientated counterclockwise). Then for any 0 ă ε ď ε 0 , we have dpω ε,1 , supp f q ą 0. Thus, from the very definition of the Cauchy transform and triangular inequality,

sup 0ăεăε 0 ż ω ε,1 |Cf ||dz| ă `8. It remains to show that sup 0ăεăε 0 ż ω ε,0 |Cf ||dz| " sup 0ăεăε 0 ż L´ ´L`ε |Cf pt `iεq|dt ă `8.
Using Lemma 3.4, we conclude that Cf P E 1 pΩ L q As mentioned above, the other ingredient in the proof of Proposition 3.2 is the following interesting result due to Hardy and Littlewood (see [START_REF] Hardy | Some properties of fractional integrals ii[END_REF]thm31], see also [START_REF] Vukotić | The isoperimetric inequality and a theorem of Hardy and Littlewood[END_REF] or [START_REF] Queffélec | Analyse complexe et applications[END_REF]Thm4.11,p. 282] for a more elementary proof).

Theorem 3.7 (Hardy-Littlewood). The Hardy space H 1 pDq embeds continuously into A 2 pDq.

We are now in a position to prove Proposition 3.2.

Proof of Proposition 3.2. In view of Lemma 3.6, we already know that Cf P E 1 pΩ L q. Now, if ϕ : D Ñ Ω L is a conformal mapping, then we will have pCf ˝ϕq ϕ 1 P H 1 pDq. From Theorem [START_REF] Hardy | Some properties of fractional integrals ii[END_REF] we obtain pCf ˝ϕq ϕ 1 P A 2 pDq, or equivalently, by simple change of variable, Cf P A 2 pΩ L q. Since Ω Ă Ω L , we obtain Cf P A 2 pΩq which is what we want to prove.

From the preceding discussions we can now deduce the claim (10). Indeed, recall from (11) that

ż Da f 1,`p zq P pzq 2 dApzq ď C ż α 1,`p Daq ´Cf γ 1,`¯p zq 2 dApzq.
Clearly, when f P L log L with compact support, the same will be true for f γ 1,( which is essentially a truncation of f composed with a rotation/translation). From Proposition 3.2 (with Ω " α 1,`p D a q being a unit square in the upper half plane with base on the real line we deduce (10) (the argument is the same for f 1,´) .

Proof of Theorem 1.3. By (8) it is enough to show that f {P P A 2 p∆q `A2 pπ ´∆q. The decomposition will be given by F 1 " pf 1,``f1,´q {P and F 2 " pf 2,``f2,´q {P . By (9) we have F 1 P A 2 p∆zD a q, and (10) implies that F 1 P A 2 pD A q. The case F 2 is treated in exactly the same way.

Proof of Theorem 1.2

As already mentioned, we have to solve the First Cousin Problem for Bergman spaces. We refer the reader to [AM04, Thm 9.4.1] where some of the ideas used below can be found. The first step into this direction is to construct a partition of unity associated with Ω " ∆ Y pπ ´∆q ":

Ω 1 Y Ω 2 .
We are thus seeking for positive smooth functions ϕ i , i " 1, 2, such that supppϕ i q Ă Ω i , ϕ 1 `ϕ2 " 1 on Ω. The existence of such a partition is of course a general fact. However, for the convenience of the reader, we give a more explicite construction of ϕ i which also allows to see the optimality of the weight of the Bergman space appearing in Theorem 1.2, at least for our method.

Lemma 4.1. Let z 0 " π 2 `i π 2 and z 1 " π 2 ´i π 2 , the upper and lower vertices of D. Then there exists χ 1 , χ 2 P C 8 pΩq such that, for i P t1, 2u,

1. 0 ď χ i ď 1, suppχ i Ă Ω i 2. χ 1 `χ2 " 1 on Ω 3. @z P D, Bχ 1 Bz pzq ď C |z´z 0 ||z´z 1 |
Proof. Let ψ P C 8 pRq be a cutoff function such that 0 ď ψ ď 1, ψ| p´8, 0s " 0, ψ| r1,`8q " 1. It is clear that ψ 1 is bounded on R. We will now create an explicit support for χ 1 , smaller than ∆zD. For that, let α : p´π 2 , π 2 q Q y Þ Ñ ´1 π py ´π 2 qpy `π 2 q and draw the curves of equations C 1 : x " αpyq `π 2 and C 2 : x " ´αpyq `π 2 . They are symmetric with respect to the line x " π 2 and are included in D (see Figure 4). Now, define χ 1 by

χ 1 px, yq " ψ ˆx `αpyq ´π 2 2αpyq ˙, x, y P ´´π 2 , π 2
and complete it by 1 on ∆zD and by 0 on pπ ´∆qzD. Let now px, yq P D. We set E for the subset of px, yq P D contained between the two curves C 1 and C 2 . Clearly, when x ă ´αpyq `π 2 , then χ 1 px, yq " 0 on the left half of DzE. Also, when x ą αpyq `π 2 , then, since αpyq ą 0 on p´π 2 , π 2 q,

x `αpyq ´π 2 2αpyq ą 1, and hence χ 1 px, yq " 1 on the right half of DzE. This implies that the function χ 1 is C 8 pΩq, it takes values between 0 and 1, is 1 on ∆zD (actually on ∆zE) and 0 on pπ ´∆qzD (actually on pπ ´∆qzE). If we write χ 2 " 1 ´χ1 , the points 1. and 2. of the lemma are verified. To obtain the last point, observe that outside E the derivatives of χ 1 vanish and if px, yq belongs to E, we have |x ´π 2 | ă |αpyq|. The point 3. follows.

It can easily be seen from the mean value theorem that we cannot hope for a better estimate than 3. in Lemma 4.1, see Remark 4.3 below.

We are now in a position to prove Theorem 1.2. Beforehand, we need to introduce an auxiliary function P pzq " 1 `z2 . Observe that multiplication by P is an isomorphic operation on A 2 `D, |pz ´z0 qpz ´z1 q| ´2˘.

Pick ϕ P A 2 `D, |pz ´z0 qpz ´z1 q| ´2˘, and set ϕ 0 " ϕP . Consider the functions h 1 " χ 2 ϕ 0 on Ω 1 h 2 " ´χ1 ϕ 0 on Ω 2 They satisfy h i P C 8 pΩ i q and ϕ 0 " h 1 ´h2 on D. To conclude we need to solve a B-problem. For that, note that Theorem 4.2. Let U be a domain in C and a ą 0. If f P L 2 loc pU q and ż U |f pzq| 2 p1 `|z| 2 q 2´a dApzq ă `8

then there exists u P L 2 loc pU q which solves the equation Bu Bz " f on U and such that a ż U |upzq| 2 p1 `|z| 2 q ´adApzq ď ż U |f pzq| 2 p1 `|z| 2 q 2´a dApzq.

We now apply this theorem with U " Ω, f " v and a " 2. More precisely, since ϕ 0 P A 2 `D, |pz ´z0 qpz ´z1 q| ´2˘a nd in view of condition 3. of Lemma 4.1, we see that v P L 2 pΩq. Then Hörmander's theorem yields a function u P L 2 pΩ, p1 `|z| 2 q ´2q solving the B-problem. Observe that this also implies that u{P P L 2 pΩq. Define now ϕ 1 " h 1 ´u on Ω 1 and ϕ 2 " h 2 ´u on Ω 2 . These are holomorphic functions since are holomorphic on Ω i . Recall also that h i P L 2 pDq and h i extends trivially to Ω i zD, so that h i {P P L 2 pΩ i q. Since u{P P L 2 pΩq, we thus get f i P A 2 pΩ i q which completes the proof of Theorem 1. Remark 4.3. Condition 3. of Lemma 4.1 is optimal in the sense that lim sup zPD,zÑz i pz ´zi q Bχ 1 Bz pzq ą 0, i " 0, 1, for any arbitrary partition of unity for tΩ 1 , Ω 2 u. This is an easy consequence of the mean value theorem. So, it does not seem possible to improve the result examinating further the First Cousin Problem here.

  Bx 2 " 0 t ą 0, x P p0, πq, ypt, 0q " u 0 ptq, ypt, πq " u π ptq t ą 0, yp0, xq " f pxq x P p0, πq, (HE)

  PW a pC `q " " f P HolpCq ˇˇˇD C ą 0, |f pzq| ď Ce π|z| and ż R |f piyq| 2 dy ă 8 * the Paley-Wiener space on the right-half plane. Then by the classical Paley-Wiener theorem,

Figure 1 :

 1 Figure 1: The square D, the path γ and the sector ∆.

Figure 2 :

 2 Figure 2: The squares D, D a and the sector ∆.

Figure 3 :

 3 Figure 3: The squares Ω et Ω L , and the path ω ε .

Figure 4 :

 4 Figure 4: Values of χ 1 on Ω, and the curves x " ˘αpyq `π{2.

  So we can define a function v P C 8 pΩq such that We need the following Hörmander L 2 -estimates for the B-equation [Hör07, Thm 4.2.1].

	Bh 1 Bz	"	Bh 2 Bz	" ´ϕ0	Bχ 1 Bz	on D.
	v "	$ ' & ' %	Bh 1 Bz Bh 2 Bz	on Ω 1 , on Ω 2 .

Proposition 3.2. Let f P L log `LpRq have compact support. Let Ω be a square in the upper-half plane one side of which is a segment I Ă R. Then the Cauchy transform Cf belongs to A 2 pΩq.