Andrea Ruiu

Bernhard Bauer-Siebenlist

Marin Senila

Thorsten Jänisch

Dominique Foix

Karine Seaudeau-Pirouley

Patrick Lacroix-Desmazes
email: patrick.lacroix-desmazes@enscm.fr

Dr P Lacroix

Promising polymer-assisted extraction of palladium from supported catalysts in supercritical carbon dioxide

Keywords: catalyst, extraction, palladium, polymer, supercritical fluid Palladium extraction in supercritical carbon dioxide

Precious metals, in particular palladium (Pd), have a wide range of daily applications, from automotive catalysts to fine chemistry production. Nevertheless, these metals are relatively rare and highly expensive, considering their massive industrial utilization. In the last decades, different recycling methods have been explored. Nowadays, the most applied methods, namely pyro-and/or hydrometallurgy, involve energy-intensive processes and/or the generation of large amounts of effluents to be treated. Thus, the development of a more sustainable recycling process of precious metals is highly desirable. In the present work, we introduce a sustainable process based on the use of a green solvent, supercritical CO2, operated under mild conditions (P = 25 MPa and T = 40 °C). The extraction process is possible thanks to the addition of CO2-soluble complexing polymers bearing pyridine units. The proposed method leads to the extraction of more than 70% of Pd from an aluminosilica-supported catalyst.

Introduction

Precious metals play a key role in our daily lives. Owing to their unique properties, these elements find applications in a wide variety of industries and products, from petrochemistry to automotive applications. The yearly demand of precious metals is increasing due to industrial applications. In particular, the demand for palladium (Pd) has risen from 242 tons in 2010 to over 305 tons in 2018 [1]. Despite the constantly growing demand of Pd over the past years, there is a supply deficit as mine production is not able to meet these demands, resulting in an increase in prices. For these reasons, alternative sources of the precious metal are necessary [START_REF] Wang | Recovery of rare and precious metals from urban mines-A review[END_REF]. To date, the recovery of precious metals from spent catalysts has been performed with different techniques [START_REF] Jadhav | A review of recovery of metals from industrial waste[END_REF]. Pyrometallurgical and hydrometallurgical technologies, vacuum carbon-thermal reduction and high temperature incineration methods are the most commonly used techniques for the recycling of precious or critical metals and are thoroughly described in various reviews [START_REF] Wang | Recovery of rare and precious metals from urban mines-A review[END_REF][START_REF] Jadhav | A review of recovery of metals from industrial waste[END_REF][START_REF] Dong | Recovery of platinum group metals from spent catalysts: A review[END_REF]. The aforementioned procedures require high temperatures (over 1000 °C) [START_REF] Peng | Pyrometallurgical Recovery of Platinum Group Metals from Spent Catalysts[END_REF] or generate large amounts of effluents (usually strong acidic or basic solutions) [START_REF] Huang | Extraction of palladium(II) from nitric acid solutions with diglycolthioamide[END_REF]. This results in energy-intensive and environmentally hazardous processes.

Recently, different methods have been studied to recover precious metals from waste materials, involving technologies based on ionic liquids [START_REF] Abbott | Processing of metals and metal oxides using ionic liquids[END_REF], polyphosphonates [START_REF] Al Hamouz | Removal of heavy metal ions using a novel cross-linked polyzwitterionic phosphonate[END_REF], ion exchange resins [START_REF] Gomes | Gold recovery with ion exchange used resins[END_REF] or solvent extraction [START_REF] Nan | Recovery of metal values from spent lithium-ion batteries with chemical deposition and solvent extraction[END_REF]. Thus, this research area is very active in the development of novel greener and safer methods to implement an improved and sustainable process to recover and recycle such precious metals.

An alternative procedure is the utilization of supercritical CO2 (scCO2) as extraction medium. scCO2 is considered a green solvent, characterized by its high availability and tunable solvent power. In addition, the supercritical domain can be achieved at mild conditions (Tc =31 °C, Pc = 7.38 MPa), making scCO2 an environmentally friendly and inexpensive medium for industrial applications. Being a nonpolar solvent, scCO2 is not effective by itself to perform metal extraction in the absence of proper additives that are able to interact with the precious metal. In the literature, dithiocarbamates, betadiketones, dithizone, and perfluorocarboxylic acids have been tested as complexing additives [START_REF] Wai | Solubility Parameters and Solubilities of Metal Dithiocarbamates in Supercritical Carbon Dioxide[END_REF][START_REF] Pereira | Recovery of indium from liquid crystal displays of discarded mobile phones using solvent extraction[END_REF][START_REF] Özel | Supercritical fluid extraction of cobalt with fluorinated and non-fluorinated β-diketones[END_REF][START_REF] Halili | Supercritical CO2 Extraction of Heavy Metals Cu, Zn and Cd from Aqueous Solution using D ithizone as Chelating Agent[END_REF][START_REF] Mochizuki | Supercritical fluid extraction of alkali metal ions using crown ethers with perfluorocarboxylic acid from aqueous solution[END_REF]. Thus, our group has recently proposed the development of a polymeric extractant soluble in scCO2, constituted by CO2-philic fragments (1,1,2,2-tetrahydroperfluorodecylacrylate, FDA) and complexing units (diphenylphosphinostyrene, DPPS) [START_REF] Li | Extraction of palladium from alumina-supported catalyst in supercritical CO 2 using functional fluorinated polymers[END_REF]. This first report showed the ability of this polymer to extract up to 50% of precious metal from an alumina-supported Pd catalyst.

In the present article, we present a novel extraction formulation to remove Pd from an aluminosilicasupported catalyst through the use of new CO2-soluble complexing polymers: poly(4-vinyl pyridinegrad-1,1,2,2-tetrahydroperfluorodecylacrylate) P(4VP-grad-FDA) and a thiol-terminated poly(4-vinyl pyridine-grad-1,1,2,2-tetrahydroperfluorodecylacrylate) P(4VP-grad-FDA)SH (Figure 1). These polymers contain pyridine complexing units able to interact with Pd species and CO2-philic FDA monomer units to confer solubility in scCO2. Although some non-fluorinated polymers such as siloxane-based [START_REF] Stoychev | Phase behavior of poly(dimethylsiloxane)-poly(ethylene oxide) amphiphilic block and graft copolymers in compressed carbon dioxide[END_REF][START_REF] Parzuchowski | Hyperbranched poly(ether-siloxane) amphiphiles of surprisingly high solubility in supercritical carbon dioxide[END_REF] or vinyl alkanoate-based [START_REF] Tapriyal | Poly(vinyl acetate), poly((1-O-(vinyloxy) ethyl-2,3,4,6-tetra-O-acetyl-β-dglucopyranoside) and amorphous poly(lactic acid) are the most CO2-soluble oxygenated hydrocarbon-based polymers[END_REF][START_REF] Zhang | Functional oligo(vinyl acetate) bearing bipyridine moieties by RAFT polymerization and extraction of metal ions in supercritical carbon dioxide[END_REF] polymers are soluble in scCO2, fluorinated FDAbased copolymers were chosen for this study because they usually exhibit exceptional solubility in dense CO2 with low sensitivity to molecular weight and nature of the chain-ends [START_REF] Lacroix-Desmazes | Macromolecular surfactants for supercritical carbon dioxide applications: Synthesis and characterization of fluorinated block copolymers prepared by nitroxide-mediated radical polymerization[END_REF][START_REF] Ma | Synthesis of hydrophilic/CO2-philic poly(ethylene oxide)-bpoly(1,1,2,2-tetrahydroperfluorodecyl acrylate) block copolymers via controlled/living radical polymerizations and their properties in liquid and supercritical CO2[END_REF][START_REF] André | Solubility of fluorinated homopolymer and block copolymer in compressed CO2[END_REF]. In contrast, non-fluorinated polymers are mostly soluble at low molecular weights and are sensitive to the nature of the chain-ends [START_REF] Shen | CO2-solubility of oligomers and polymers that contain the carbonyl group[END_REF][START_REF] Hu | Effect of molecular weight on CO2-philicity of poly(vinyl acetate) with different molecular chain structure[END_REF]. Moreover, we have focused on gradient copolymers instead of block copolymers because the latter usually require higher CO2 pressures to be solubilized in dense CO2 [START_REF] Ribaut | Solubility and Self-Assembly of Amphiphilic Gradient and Block Copolymers in Supercritical CO 2[END_REF]. This method is proposed as an alternative and innovative process to extract Pd from supported catalysts by using a green method, avoiding the use of high temperatures and the generation of hazardous effluents.

Experimental Section

The catalysts used for these studies have been provided by Heraeus Deutschland GmbH & Co. KG: pristine catalyst D (CAT D), catalyst D after reduction over 4h under H2 atmosphere at 500 °C (CAT Dred), and catalyst D after reduction and subsequent oxidation over 4h under Cl2 atmosphere at 470 °C (CAT D-ox) (Figure 2). CAT D is a pristine, industrial aluminosilica-supported catalyst used for the initial extraction experiments. This catalyst is similar to the common spent catalysts that Pd would normally be recycled from in terms of material and catalyst type, and therefore would help to reflect realistically on the performance of the extraction in industrial settings. The characterizations of the catalysts are reported in the supporting information (inductively coupled plasma optical emission spectrometry (ICP-OES), X-ray photoelectron spectroscopy (XPS), transmission electron microscopy (TEM), scanning electron microscopy with energy dispersive X-ray spectroscopy (SEM-EDX), nitrogen adsorptiondesorption isotherms (BET)). Carbon dioxide (CO2, SFE 5.2, Air Liquide, 99.9%) was used as received. The polymer synthesis has been performed following the procedures shown in the Supporting Information (SI), adapted from a previous work [START_REF] Ribaut | Synthesis of gradient copolymers with complexing groups by RAFT polymerization and their solubility in supercritical CO 2[END_REF]. The polymer characterization is also shown in the SI. The procedure for the extraction was as follows (Figure S1): catalyst (≈ 200 mg) and P(4VP-grad-FDA) (≈ 220 mg) were added in the 35 mL stainless-steel extraction cell (Top Industrie, France), which was then tightly closed. In the case of CAT D-ox, 0.02 mL of DIPEA (N,N-diisopropylethylamine) were added to buffer the eventual release of HCl during the extraction (chlorine in Na2PdCl4). The ISCO pump was stabilized at P = 27 MPa and T = 35 °C. Afterwards, the extraction cell was filled with CO2 until 25 MPa at 40 °C were reached in the cell. The extraction was performed under magnetic stirring at 100 rpm for one hour at 25 MPa and 40 °C (batch conditions), and then the cell was flushed with ≈160 ml of CO2 (26 MPa and 35°C in ISCO pump) at a flow rate of approximately 0.6-1.2 mL/min. Afterwards, the extraction cell was opened and the catalyst was recovered (Sample A). The cell was cleaned with acetone, which was subsequently collected and then evaporated (Sample B). Samples A and B were analysed by ICP-OES to determine the amount of precious metal extracted: Extraction conversion (%) = E (%) = [1 -(mass of PdsampleA + mass of PdsampleB)/(mass of Pdinitial catalyst)]×100. As detailed in Table S1, the calculated extraction conversion (E %) is related to the total mass of Pd (samples A and B) left in the cell after extraction, therefore there is no further correction to apply. The errors bars E/E (%) on extraction conversion have been calculated considering the following accuracies: accuracy of weighing balance of ±1 mg, ICP-OES measurements are accurate within ±6 %.

Results and discussion

The catalysts used for these extraction experiments (CAT D, CAT D-red and CAT D-ox) were characterized through different analytical techniques to determine the concentration of the supported Pd (ICP-OES) and its oxidation state (XPS), the porosity of the catalyst (BET), the spatial distribution of the metal in the catalysts (SEM-EDX), and the metal particle size (TEM) (see the Supporting Information). All the catalysts have 2 wt% of Pd (Tables S2-S3) and they are supported on an aluminosilica support (Table S4). For all the studied catalysts, the metal is present mainly on the surface of the support (Figures S2-S3); nonetheless, traces of metal could also be detected in the inner part of the material (Figure S4). The average pore size of the aluminosilica support is 21-22 nm, determined by nitrogen adsorption-desorption isotherms (BET) (Figures S5-S7, Table S5). Previous studies [START_REF] Ribaut | Solubility and Self-Assembly of Amphiphilic Gradient and Block Copolymers in Supercritical CO 2[END_REF][START_REF] Ribaut | SANS study of the self-organization of gradient copolymers with ligand groups in supercritical CO2[END_REF] have shown that amphiphilic fluorinated polymers are prone to form micelles in scCO2. In particular, polymers weighing between 10 and 20 kg/mol form micelles, with diameters in the order of 10 nm. Thus, the pore diameter of the catalyst support is larger than the diameter of the micelles, allowing a proper interaction of the polymer with the precious metal even in the inner parts of the supported catalyst. All the supported catalysts used in this study have similar physical properties in terms of pore size and metal distribution. The main differences between the catalysts are the Pd oxidation states and the nature of the Pd species on the support, as well as the Pd particle size. CAT D contains palladium oxide PdO (100%) (Figure S8), with an average particle size of 2.7 nm (Figure S9). CAT D-red contains Pd 0 (79%) and PdO (21%) (Figure S10), characterized by an increase of the average particle size, up to 3.3 nm (Figure S11). Finally, CAT D-ox contains mainly Na2PdCl4 (85%) with a minor amount of PdO (15%) (Figures S12-S13, Table S4). Consistently, in the case of CAT D-ox, TEM analysis showed the presence of a smaller amount of Pd particles at nanoscale dimensions, corresponding to the remaining PdO (Figure S14). Indeed, due to the low contrast of the Na2PdCl4 in TEM [START_REF] Hong | Selective Heterogeneous Catalytic Hydrogenation by Recyclable Poly(allylamine) Gel-Supported Palladium(0) Nanoparticles[END_REF], it was not possible to see this salt which was adsorbed on the catalyst.

The variation of the Pd oxidation state and the nature of the Pd species in the different catalysts are due to a pretreatment applied for the purpose of this work (Figure 2). The aim of modifying the oxidation state and the nature of the Pd species is to evaluate the efficiency of the extraction process on different Pd species (Pd(II)O, Pd°, and Na2Pd(II)Cl4) while keeping the same catalyst support (aluminosilicate) and metal distribution.

The polymers were synthesized by controlled radical polymerization using the reversible additionfragmentation chain transfer (RAFT) technique [START_REF] Ribaut | Synthesis of gradient copolymers with complexing groups by RAFT polymerization and their solubility in supercritical CO 2[END_REF]. The Alfrey and Price Q and e values of 4VP and FDA are available in the literature: Q4VP=2.47 and e4VP=0.84 [START_REF] Odian | Chain Polymerization[END_REF], QFDA=0.44 and eFDA=0.45 [START_REF] Guyot | Détermination des coefficients d'Alfrey et Price de monomères acryliques fluorés et du méthacrylate de morpholinoéthyle[END_REF]. From these Q and e values, the reactivity ratios could be estimated by using the classical Alfrey and Price equations [START_REF] Odian | Chain Polymerization[END_REF]: r4VP=4.05 and rFDA=0.21. Thus, 4VP is preferably introduced during copolymerization with FDA and there is a composition drift within the polymer chains (the beginning of the polymer chain is enriched in 4VP units), leading to a gradient copolymer structure for P(4VP-grad-FDA) (Figure S15) and P(4VP-grad-FDA)SH (Figure S16). The cloud point curves of the copolymers were determined in dense CO2 (Figure 3, Table S6). The copolymers were shown to be soluble in mild conditions of pressure, lower than 30 MPa in the full range of temperatures from 25 to 65 °C, as expected from previous works [START_REF] Ribaut | Solubility and Self-Assembly of Amphiphilic Gradient and Block Copolymers in Supercritical CO 2[END_REF][START_REF] Ribaut | Synthesis of gradient copolymers with complexing groups by RAFT polymerization and their solubility in supercritical CO 2[END_REF][START_REF] Ribaut | SANS study of the self-organization of gradient copolymers with ligand groups in supercritical CO2[END_REF]. Notably, the aminolyzed copolymer P(4VP-grad-FDA)SH (Figure S16) has very similar cloud point pressures as its parent copolymer, indicating the negligible effect of the chain-end on the cloud point curve thanks to the high solubility of this kind of FDA-based fluorinated copolymer in dense CO2. The extraction procedure involves the use of mild supercritical operating conditions, at P = 25 MPa and T = 40 °C, similar to those currently used for the decaffeination in scCO2 at industrial scale [START_REF] Lack | Commercial scale decaffeination of coffee and tea using supercritical CO2[END_REF], making the metal extraction process potentially readily available for upscaling and commercial applications. Indeed, Manjare et al. [START_REF] Manjare | Supercritical fluids in separation and purification: A review[END_REF] have reported that the relatively higher investment requirement in supercritical fluid extraction (SFE) processes is well compensated by several benefits, such as the low cost of CO2, shorter batch times, and negligible energy requirement of solvent recovery from the extract. In addition, there is no cost of pollution treatment since there is practically no effluent generated by SFE plants. In the present work, the conditions of temperature and pressure (P = 25 MPa and T = 40 °C) were chosen above the cloud point curve of the copolymers, ensuring that the copolymers are solubilized in scCO2 during the extraction process. The goal of the first experiments was to verify that the nonpolar scCO2 alone is unable to extract Pd. Without polymer, for all the tested catalysts, less than 3% of Pd was removed from the catalyst support (Figure 4) (Table 1, runs E1, E4, E7), confirming the negligible solubility of the Pd species in neat scCO2 and the requirement of a complexing agent as additive. Subsequently, the extractions of Pd from the supported catalysts have been performed in scCO2 in the presence of P(4VP-grad-FDA) as complexing agent (Figure 1). As can be seen from the extraction tests with CO2 alone (E1, E4, E7), the polymer is fundamental for the metal extraction. This was expected as supercritical CO2 is a nonpolar system and consequently there is almost no chemical interaction between the metal and CO2. Therefore, it was decided to prepare a CO2-philic polymer bearing several pyridine units, well-known to interact and form complexes with metals like Pd and Pt [START_REF] Matlock | A pyridine-thiol ligand with multiple bonding sites for heavy metal precipitation[END_REF][START_REF] Tu | A robust hydrophilic pyridine-bridged bis-benzimidazolylidene palladium pincer complex: Synthesis and its catalytic application towards Suzuki-Miyaura couplings in aqueous solvents[END_REF][START_REF] Newkome | Chemistry of heterocyclic compounds. 81. Palladium(II) complexes of pyridine-and pyrazine-based ligands with trans bis(carbon-metal) bonds. Ligand synthesis, complexation, and crystal structure[END_REF].

The first polymer-assisted extraction experiments were performed at 25 MPa and 40 °C on CAT D, CAT D-red and CAT D-ox with the polymer P(4VP-grad-FDA) (Table 1, runs E2, E5, E8). This complexing polymer (Mn 11830 g/mol) contains an average of 20 units of pyridine, allowing for good complexing ability, and 18 FDA units (Figure S15), necessary to obtain a polymer that is soluble in scCO2 under mild conditions of pressure. The extraction on CAT D showed low efficiency (19.8%) (Figure 4 and Table 1, run E2), due to the strong interaction between PdO and the aluminosilica support as well as poor reactivity of such Pd derivatives toward organic ligands. With the reduced catalyst (Cat D-red) containing Pd 0 , the extraction was only slightly improved (24.1%) (Figure 4, Table 1, run E5). The low efficiency of this polymer to interact with and remove the metal from the supported catalyst can be explained by the low affinity of the pyridine units as ligand in relation with the Pd 0 nanoparticles. As shown in the Supporting Information, for both PdO and Pd 0 loaded catalysts, the Pd nanoparticles on the aluminosilica support have a diameter in the order of 3 nm. Presumably, the polar pyridine units favor the adsorption of the amphiphilic gradient copolymer on the surface of the Pd nanoparticles. Thus, thanks to the CO2-solubility of the polymer due to the FDA units, the PdO and Pd° nanoparticles can be dispersed in scCO2, with the polymer playing the role of a steric stabilizer (Figure 5a). It is important to notice the influence of the complexing polymer on the extraction process: in fact, the extraction of Pd increases from 0.6% and 3.2% (respectively for CAT D and CAT D-red without polymer) to 19.8% and 24.1% in the presence of the amphiphilic CO2-soluble polymer P(4VP-grad-FDA) (Figure 4).

Finally, the polymer-assisted extraction procedure with the polymer P(4VP-grad-FDA) has been tested with the catalyst CAT D-ox, which contains mainly Na2PdCl4 (Figure 4 and Table 1, run E8). The extraction experiment led to a remarkable efficiency, allowing for the removal of 73.3% of the Pd absorbed on the catalyst. The successful Pd extraction can actually be observed visually by a change of color of the supported catalyst (Figure 6). As illustrated in Figure 2, the colored Pd species give the reddish-brown color to CAT D-ox (the aluminosilica support is white). In Figure 6, the reddish-brown color of CAT D-ox decreased after extraction because of the removal of the colored Pd species from the catalyst support. There was no significant volume nor mass change of the supported catalyst after extraction because there is only 2 wt% Pd on the catalyst support. The improved result obtained with CAT D-ox can be explained by the enhanced reactivity of the Na2PdCl4 halogenated species with the pyridine ligand in comparison with the oxide (PdO) (CAT D) or metallic (Pd°) (CAT D-red) species. Indeed, halogenated Pd derivatives are commonly used as precursors in organometallic chemistry to synthesize novel complexes containing the precious metal [START_REF] Dey | Pyridine-and 3-/6-Methylpyridine-2-tellurolate Complexes of Palladium(II) and Platinum(II)[END_REF][START_REF] Batten | Some Coordination Chemistry of the Bidentate Nitrogen-Donor Ligand 2-(2-Aminophenyl)pyridine, Artic[END_REF][START_REF] Ye | Palladium nanoparticles embedded chitosan/poly(vinyl alcohol) composite nanofibers as an efficient and stable heterogeneous catalyst for Heck reaction[END_REF][START_REF] Solodenko | Microwave-Assisted Suzuki-Miyaura Reactions with an Insoluble Pyridine-Aldoxime Pd-Catalyst[END_REF][START_REF] Ha | A second monoclinic polymorph of (pyridine-2-carboxaldehyde oximato-κ 2 N , N ′)(pyridine-2-carboxaldehyde oxime-κ 2 N , N ′)palladium(II) chloride[END_REF]. Thus, the complexation of the precious metal by the polymer ligand sites, combined with the CO2-solubility of the polymer, makes the solubilization of the Pd/polymer supramolecular species and their transportation in the scCO2 medium possible, resulting in a promising polymer-assisted extraction of Pd from the aluminosilicasupport of the catalyst (Figure 5b). After these first experiments, the polymer P(4VP-grad-FDA) has been modified by aminolysis in order to obtain P(4VP-grad-FDA)SH (Figure 1), bearing a thiol end-group, thus providing a good Pdcomplexing unit to the polymer. This functionalization was performed to improve the metal extraction. Indeed, the sulphur-based molecules are well-known ligands for Pd species and are commonly used as complexing agents for metal recovery [START_REF] Ko | Selective removal of heavy metal ions by disulfide linked polymer networks[END_REF][START_REF] Iwao | Recovery of palladium from spent catalyst with supercritical CO2 and chelating agent[END_REF][START_REF] Suoranta | Recovery of palladium, platinum, rhodium and ruthenium from catalyst materials using microwave-assisted leaching and cloud point extraction[END_REF].

Color before

Color after [a] General conditions: mcatalyst = 200 mg, mpolymer = 220 mg , mCO2,batch step = 35 g, mCO2,flushing step = 145 g. [b] Determined by inductively coupled plasma -optical emission spectrometry (ICP-OES).

Unexpectedly, the extractions were not improved by the use of the thiol-terminated P(4VP-grad-FDA)SH polymer as complexing agent (Figure 4, Table 1, runs E3, E6, E9). Therefore, the thiol group of P(4VP-grad-FDA)SH does not bring benefits to the steric stabilization of PdO (CAT D) and Pd° (CAT Dred) nanoparticles, nor in the supramolecular complexation of Na2PdCl4 (CAT D-ox). In all cases, the contribution of the single thiol group is not significant enough compared to the 20 4VP units of the polymer chain.

The extraction conversion of 73% obtained for CAT D-ox with P(4VP-grad-FDA) copolymer represents a promising polymer-assisted extraction of palladium from supported catalysts in scCO2. Indeed, even though the experimental conditions for the extraction have not been optimized yet, high extraction conversions have already been accomplished. Tan et al. [START_REF] Zhang | Functional oligo(vinyl acetate) bearing bipyridine moieties by RAFT polymerization and extraction of metal ions in supercritical carbon dioxide[END_REF] have reported high extraction conversions (>70%) of Ni 2+ , Co 2+ and Cu 2+ by supercritical CO2 extraction assisted by oligo(vinyl acetate) bearing a bipyridine moiety, similar to those reported by our group [START_REF] Chirat | Extraction of Cobalt Ion from Textile Using a Complexing Macromolecular Surfactant in Supercritical Carbon Dioxide[END_REF] for cobalt extraction using a fluorinated polymer bearing phosphonic acid moieties; nevertheless, the support of the metal was not aluminosilica but filter paper [START_REF] Zhang | Functional oligo(vinyl acetate) bearing bipyridine moieties by RAFT polymerization and extraction of metal ions in supercritical carbon dioxide[END_REF] or cotton fabric, [START_REF] Chirat | Extraction of Cobalt Ion from Textile Using a Complexing Macromolecular Surfactant in Supercritical Carbon Dioxide[END_REF] respectively. In previous works of Pd extraction from supported catalysts in scCO2, even higher extraction conversions (up to 99%) were obtained, but under optimized conditions and using hazardous chemicals like HNO3 [START_REF] Iwao | Recovery of palladium from spent catalyst with supercritical CO2 and chelating agent[END_REF][START_REF] Wang | Dissolution of precious metals in supercritical carbon dioxide[END_REF][START_REF] Faisal | Recovery of precious metals from spent automobile catalytic converters using supercritical carbon dioxide[END_REF]. Compared to these earlier studies, the Pd extraction conversion of 73% obtained in the present work, without optimization and without generating hazardous waste streams, appears to be a very promising result.

The amount of polymer used in run E8, leading to 73 % extraction conversion, is 0.259 g of P(4VPgrad-FDA) for 0.208 g of supported catalyst CAT D-ox. This weight ratio of polymer/catalyst of 1.2 is quite low compared to the amount of solvent used in hydrometallurgy where the liquid/solid ratio for leaching is typically in the range of 10[48]. The extraction methods such as pyrometallurgical processes and various hydrometallurgical and leaching methods [START_REF] Ding | An Efficient Leaching of Palladium from Spent Catalysts through Oxidation with Fe(III)[END_REF] have been reviewed for the recovery of precious metals from supported catalysts [START_REF] Saguru | A review of recent studies into hydrometallurgical methods for recovering PGMs from used catalytic converters[END_REF][START_REF] Ding | Recovery of precious metals from electronic waste and spent catalysts: A review[END_REF]. Among them, alternative methods such as supercritical fluids extraction (scCO2 and supercritical water), ionic liquid-assisted extraction [START_REF] Firmansyah | Selective recovery of platinum group metals from spent automotive catalysts by leaching and solvent extraction[END_REF], ultrasound-assisted pre-treatments for bioleaching [START_REF] Karim | Ultrasound-assisted nitric acid pretreatment for enhanced biorecovery of platinum group metals from spent automotive catalyst[END_REF] and microwave-assisted leaching [START_REF] Suoranta | Recovery of palladium, platinum, rhodium and ruthenium from catalyst materials using microwave-assisted leaching and cloud point extraction[END_REF] are considered to also have good potential to leach precious metals such as Pd. Nevertheless, there are still some challenges to achieving environment-friendly and sustainable recycling for precious metals with a high recovery rate. Thus, future technologies for precious metals recycling need to be more efficient and to have less environmental impact, lower cost, and higher recovery rate. In this context, polymer-assisted extraction in supercritical CO2 could be a way to avoid the use of low molecular weight, hazardous leaching agents.

As a conclusion, a promising method for the extraction of Pd from aluminosilica-supported catalysts in scCO2 under mild conditions (P = 25 MPa and T = 40 °C) has been demonstrated by using a CO2soluble complexing polymer containing pyridine units P(4VP-grad-FDA). In this work, over 70% of the Pd precious metal has been removed from an aluminosilica support, by an environmentally friendly and non-destructive process, avoiding high energy demanding procedures or generation of polluting effluents. These studies will be pursued to further improve this extraction process, for instance through the use of different CO2-soluble complexing polymers (including non-fluorinated ones) and testing different extraction parameters (time, pressure, temperature). This method opens up the route for the development of a sustainable technology able to extract precious metals from porous supports, the first step of a recycling process with an increasing importance in modern industrial activities and in the emerging circular economy. Further steps will include the separation of Pd from the polymer-Pd mixture and the investigation of various methods for Pd recovery after its extraction by scCO2. In addition, a comparative life cycle assessment (LCA) [START_REF] Marco | Life cycle assessment of supercritical CO2 extraction of caffeine from coffee beans[END_REF][START_REF] Ghodrat | A Comparative Life Cycle Assessment of[END_REF][56] between a conventional process and the process reported herein is envisioned to evaluate the technico-economical feasibility and the sustainability of the proposed polymer-assisted extraction process of precious metals in supercritical CO2 compared to other traditional methods.

Recycling

Figure 1 .

 1 Figure 1. Structures of the gradient copolymers P(4VP-grad-FDA) and P(4VP-grad-FDA)SH used for the extractions of Pd from the supported catalysts.

Figure 2 .

 2 Figure 2. Pictures of the Pd supported catalysts CAT D, CAT D-red and CAT D-ox studied in this work.

Figure 3 .

 3 Figure 3. Cloud point curves of P(4VP-grad-FDA) (diamond) and P(4VP-grad-FDA)SH (square) in dense CO2 (at a polymer concentration of ca. 1 wt% of polymer relative to CO2).

Figure 4 .

 4 Figure 4. Extractions of palladium from CAT D (E1, E2, E3) , CAT D-red (E4, E5, E6) and CAT D-ox (E7, E8, E9) in the absence of complexing polymer (E1, E4, E7) and in the presence of complexing polymers P(4VP-grad-FDA) (E2, E5, E8) and P(4VP-grad-FDA)SH (E3, E6, E9) in scCO2 at T = 40 °C and P = 25 MPa.

Figure 5 .

 5 Figure 5. (a) Steric stabilization of the PdO or Pd° metallic nanoparticles by the polymer P(4VP-grad-FDA) in scCO2 for the catalysts CAT D and CAT D-red; (b) formation of CO2-soluble supramolecular complexes Pd/P(4VP-grad-FDA) for the catalyst CAT D-ox.

Figure 6 .

 6 Figure 6. Change of color of the supported catalyst CAT D-ox after supercritical CO2 extraction of Pd assisted by the copolymer P(4VP-grad-FDA) at 40 °C and 25 MPa: a) color of CAT D-ox before extraction; b) color CAT D-ox after extraction.

Table 1 .

 1 Polymer-assisted extractions of Pd from aluminosilica-supported catalysts in supercritical CO2 at T = 40 °C and P = 25 MPa[a] .

							Extracted
							Pd from
	Run	Supported catalyst	Polymer	Complexing group	Polymer/Pd molar ratio	Complexing group/Pd molar ratio	the support of the catalyst
							(%) [b]
	E1		None	None	0	0	0.6 ± 0.6
	E2	CAT D	P(4VP18-grad-FDA20)	4VP	0.56	11.1	19.8 ± 2.9
	E3		P(4VP18-grad-FDA20)SH	4VP SH	0.50	10.5	16.6 ± 2.6
	E4		None	none	0	0	3.2 ± 0.9
	E5	CAT D-red	P(4VP18-grad-FDA20)	4VP	0.55	11.1	24.1 ± 3.5
	E6		P(4VP18-grad-FDA20)SH	4VP SH	0.50	10.6	24.8 ± 3.6
	E7		None	none	0	0	1.5 ± 0.7
	E8 E9	CAT D-ox	P(4VP18-grad-FDA20) P(4VP18-grad-FDA20)SH	4VP 4VP SH	0.56 0.51	11.2 10.7	73.3 ± 9.6 67.6 ± 8.9

  the Platinum Group Metals from Automobile Catalytic Converter: An Australian Perspective, Metall. Mater. Trans. E. 4 (2017) 77-88. doi:10.1007/s40553-017-0109-1. [56] M.N. Le, M.S. Lee, A Review on Hydrometallurgical Processes for the Recovery of Valuable Metals from Spent Catalysts and Life Cycle Analysis Perspective, Miner. Process. Extr. Metall. Rev. 00 (2020) 1-20. doi:10.1080/08827508.2020.1726914.

Acknowledgements

The authors thank ANR (ANR-17-MIN2-0004-01), ADEME, Federal Ministry of Education and Research (grants number 033RU009A; 033RU009B), CCCDI-UEFISCDI (project number COFUND-ERANET-ERAMIN-SUPERMET, contract 48/2018) for financial support of the SUPERMET project


Meher Azouzi
email: meher.azouzi@uha.fr

Ated Ben Khalifa
email: ated.bkh@gmail.com

Anne Collaine
email: anne.collaine@uha.fr

Michel Tourlonias
email: michel.tourlonias@uha.fr

Characterization of the area roughness of a GFRP by a 3D profilometer after trimming

Keywords: GFRP, Composite Trimming, Area Roughness, 3D Profilometer, Central composite design, MSR

The aim of this study is to characterize the link between the cutting conditions used while trimming a composite material glass fiber reinforced polymer (GFRP) and the surface quality by characterizing the area roughness of the machined surfaces. In the experimental evaluation, a central composite design with 20 combinations was used to study cutting parameters (cutting speed (V c ), radial engagement (a e ) and tooth feed (f z )). The area roughness parameters (Sa, Sq, Sz) were measured by a KEYENCE VHX-6000 3D profilometer. Response Surface Methodology (MSR) were used to determine mathematical models using experimental data.

Introduction

The composite materials industry is a sector that has made a great deal of progress over many years [START_REF]Les composites : des matériaux d'avenir -Partie 2 : Marché et évolutions[END_REF] [START_REF]2016: a good year for composites[END_REF]. Today, the manufacturing of this material is mainly divided between glass fiber reinforced composites (GFRP) and carbon fiber reinforced composites (CFRP). Although they are less efficient, GFRP composites now represent 95% by volume of the composites produced in Europe [START_REF] Witten | Composites Market Report 2016 -Market developments, trends, outlook and challenges[END_REF]. The process of obtaining a composite material part involves two stages: a first stage consists of manufacturing and a second stage called finishing. Trimming is one of the most common finishing processes used. Obtaining sufficient quality in terms of dimensional accuracy, surface finish and absence of residual fibers are the main difficulty in machining composites while at the same time assuring a certain productivity: speed of production, tool costs, etc... In view of the difficulties in obtaining finished quality workpieces, various studies are interested in proposing evolutions of the machining finishing process in order to improve the results [START_REF] Mehbudi | Applying Ultrasonic Vibration to Decrease Drilling-Induced Delamination in GFRP Laminates[END_REF] [START_REF] Uhlmann | Machining of Carbon and Glass Fibre Reinforced Composites[END_REF]. The combined effect of cutting parameters on surface quality or tool degradation in GFRP composite drilling operations has been the subject of extensive study [6] [7] [8] [START_REF] Palanikumar | Experimental investigation and optimisation in drilling of GFRP composites[END_REF]. In the case of milling GFRP composites, the influence on the surface quality of the machining direction in relation to the fiber orientation is known [START_REF] Venu Gopala Rao | Machining of UD-GFRP composites chip formation mechanism[END_REF].

The same can also be said for the influence of cutting parameters [START_REF] Parida | Optimization of Cutting Parameters for Surface Roughness in Machining of GFRP Composites with Graphite/fly Ash Filler[END_REF]. However, several research works haven't shown a complete and global study on the trimming of GFRP composites combining tool choice and parameters choice and results, both in terms of quality and productivity. The definition of behaviors, generic theoretical principles are studied but the experiments carried out concern carbon fiber reinforced composites [START_REF] Boudelier | Investigation of CFRP machining with diamond abrasive cutters[END_REF] [13]. In the present work, the roughness of the machined surface states is investigated. Indeed, we are interested in the use of area roughness parameters to describe the quality of the machined surface.

Response Surface Methodology

The principle is to model the experimental response surface. Among the many types of plans to build response surfaces, we will use a central composite design. Indeed, this technique has the advantage of being suitable for the sequential conduct of a study and requiring a relatively few experiments [START_REF] Box | Empirical model building and response surface[END_REF]. RSM allows to evaluate not only the linear effects but also quadratic effects as well as the effects of interaction between the different operating variables, the model used is the quadratic model with interactions:

2 0 i i ij i j j j ij Y a a X b X X c X = + + +    (1)
Where a 0 , a i , b ij and c j are the regression coefficients of the model and X i , X j are the explanatory variables of the model.

Material and methods

Material

The composite material was made with Vetrotex 136 tex continuous glass fibers, reference ZTW EC13 136 tex TD22C with a diameter of 13 μm. The matrix is initially liquid thermoset matrix with two components: epoxy / amine system CY219 and Renlam HY5160 (Huntsmann). The test pieces are made by a method of filament winding to obtain a unidirectional composite. The equipment of the manufacturing composite plates is shown in fig. 1. Indeed, the fiberglass filament is wound on a parallelepiped mold animated by a rotational movement managed by a motor whose speed is controlled. The filament slides in a guiding system whose movement is controlled by the rotation of a worm drive, and is driven by the rotation of the mold. The combination of the rotational speed of the mold and that of the worm drive ensures the implantation of the glass fibers on the mold in the form of parallel lines with two possible orientations: 0 ° and 90 °. The rotation of the worm drive is reversed at each end. At each inversion, the system is stopped manually to reassure manual impregnation with the resin.

Machine and tool used

The machining tests were carried out on a Charly 2U machine equipped with a spindle turning up to 24000 𝑡𝑟 / 𝑚𝑖𝑛, allowing feed rates up to 100 𝑚𝑚 / 𝑠 and with a maximum spindle power of 1,1 𝑘𝑊. For the cutting tool (Fig. 2), our choice fell on a one-piece SECO carbide milling cutter with a diameter 𝑑 = 6 𝑚𝑚 whose main characteristics are shown in table 1.

Machining strategy

During machining the composite plate is mounted between two flanges tightened by 8 screws (Fig. 3). During the trimming operation, the machining strategy adopted is climb milling. Indeed, the tool is driven by a translational movement with a feedrate V f (mm / min) and a rotation speed N (tr/min). As part of this work, all the tests were carried out with a tool overhang D ep = 1mm and a width of plate a p =4 mm. The milling cutter goes out of the spindle about 25 mm.

Design of experiments

A central composite design with 20 combinations (Fig. 4) was chosen to study the influence of the three selected cutting parameters (cutting speed V c , radial engagement a e and tooth feed f z ). The values of parameters were established in accordance with the recommendations provided by the SECO cutting tool manufacturer. 2, where:

4 N  =
and N is the number of fractional plan tests. Here N = 8, so: 4

Analysis method of area roughness

We have at our disposal a 3D KEYENCE VHX-6000 digital microscope that will allow us to measure area roughness parameters. This microscope is essentially characterized by: advanced lighting and focus, multiple illumination and direct depth composition. With digital microscope, the operator can tilt the lens and camera at a maximum angle of 90 degrees, eliminating the need to manipulate the object by hand and thus facilitate control. The XY stage allows a displacement range of 100 × 100 𝑚𝑚 so observation and image assembly are faster. For magnification we can go from 20 × to 2000 × with one lens. The aim is to characterize the machined surface with area roughness parameters. The microscope used allows us to scan the entire machined surface to calculate the desired parameters. The scanning operation of the entire surface to do the 3D reconstruction is only done if the lens and the camera are perpendicular. In our case we cannot satisfy this condition because of the length of the machined samples. For this reason, we have designed a support inclined at 60 ° to the vertical to measure all the machined plates. In order to properly characterize the machined surface, we chose to use a 1000 × magnification and to average 8 evenly distributed measurement points (Fig. 5). For each "point" we defined a square measurement zone of 250 μ𝑚 of side and recorded the values of Sa, Sq and Sz by setting the filters to the following values: S-Filter = 2 μm, L-Filter = 0,25 mm. This strategy is based on the instructions in ISO 25178-2.

Results and discussion

The analysis of the machined surface state according to the experimental procedure detailed above, allowed us to recover several area roughness parameters. Among these parameters, the choice was fixed on two parameters which have a mean of measurement which are Sa and Sq and a parameter which describes a maximum height which is Sz. Indeed, these parameters are defined as follows: (4)

Results

The different results of the area roughness parameters are presented in the following table (Table 3).

The recovered results show a coincidence at the level of the trend of evolution for the Sa and Sq responses and this one is entirely logical considering the similarity of these two parameters. The found values of the area roughness are in line with the recommendations of the manufacturer VHX.

For the area roughness response Sz, it has the same trend of evolution as the area roughness response Sa with a difference in values as shown fig. 6.

(a) (b)

Results study with "Minitab"

For the study of surface roughness results with Minitab, we will limit ourselves only to the study of the area roughness response Sa.

Analysis of the empirical model

The studied model takes into account the linear, interaction and quadratic effects. The regression coefficients of this model make it possible to express the relation of the area roughness Sa according to the three parameters studied: The fig. 7 shows that the tooth feed-tooth feed interaction is the most influential. It is followed by radial engagement, cutting speed and tooth feed.

S V a f V V a a f f V a V f a f = + + - -  -  +  -  +  -  (5)

Analysis of response surfaces

The relationship between the different operating variables and the response studied is illustrated in the three-dimensional representations of the response surfaces:

• The Fig. 8 (b) illustrates the effect of the interaction (V c , f z ) on the area roughness Sa, the minimum value of Sa corresponds to the central values of V c and f z . • The Fig. 8 (c) illustrates the effect of the interaction (a e , f z ) on the area roughness Sa, the influence of a e is more important than f z .

Conclusions and perspectives

In this work, various GFRP composite trimming tests were presented in order to study the influence of the cutting parameters V c , a e and f z on the quality of the surfaces obtained. From the study of the surface quality and the results of the experimental designs we can conclude that the machined surface roughness is highly sensitive to the variation of the cutting parameters.

Many approaches can be considered to progress in this work, such as improving the experimental protocol, testing other tool engagements (small pass depth and high radial engagement), investigating larger variation ranges of f z and V c , testing parameter variations to see if the influence is the same, completing the study with work on tool lifespan ...

While the first results are encouraging and the prospects numerous, they highlight the need to quantify vibrations and to have at our disposal a means of characterizing the vibratory behavior of the workpiece during trimming. Vibration measurement will help to better define the operating ranges of a tool in relation to cutting conditions.

Fig. 1 .

 1 Fig. 1. Device for manufacturing composite plates

Fig. 2 .

 2 Fig. 2. Cutting tool

Fig. 3 .

 3 Fig. 3. Trimming strategy

Fig. 4 .

 4 Fig. 4. Central composite design The experimental design consists of 20 tests: 8 tests corresponding to the orthogonal plane, 6 tests for the axial points and 6 tests at the central point (0, 0, 0) to check the repeatability. The domain of variation of the coded factors is brought back to the interval [Min; Max] divided into 5 levels: -δ, -1, 0, 1, +δ which are calculated according to

Fig. 5 .

 5 Fig. 5. (a) Area roughness measurement instrument, (b) Area roughness measurement strategy

Fig. 6 .

 6 Fig. 6. Variation of the area roughness parameters Sa, Sq and Sz

Fig. 7 .

 7 Fig. 7. Pareto diagram of the area roughness response Sa

Fig. 8 .

 8 Fig. 8. Response surface results: (a) fz=0,006 mm/tooth, (b) ae=0,5 mm, (c) Vc=170 m/min

Table 1 .

 1 Characteristics of the tool used

	number Manufacturer's reference Teeth	Coating	Specifications
	840060R050Z4.0-DURA	4	DURA	Double helix to avoid delamination

Table 2 .

 2 Variation of the cutting parameters and their levels

							8 1, 68  ==	.
	Levels					Real variables			Vc (m/min) ae (mm) fz (mm/tooth)
	-δ (-1,68)							Min						130	0,1	0,04
	-1	Min	2 +	Max	-	1 ( 	Max	-	Min	2 +	Max	)	146	0,26	0,05
	0						Min	+	Max				170	0,5	0,06
									2
	1	Min	2 +	Max	+	1 ( 	Max	-	Min	2 +	Max	)	194	0,74	0,07
	+δ (1,68)							Max						210	0,9	0,08

Table 3 .

 3 Experimental matrix and area roughness responses

	Test no. Vc (m/min) ae (mm) fz (mm/tooth)	Sa (μm)	Sq (μm)	Sz (μm)
	1	146	0,26	0,05	1,9	2,9	26,2
	2	194	0,26	0,05	2,2	3,6	33,0
	3	146	0,74	0,05	2,7	4,1	34,5
	4	194	0,74	0,05	2,5	3,9	37,5
	5	146	0,26	0,07	2,8	4,1	37,2
	6	194	0,26	0,07	3,1	4,5	34,9
	7	146	0,74	0,07	3,1	4,8	42,8
	8	194	0,74	0,07	3,1	4,7	41,0
	9	130	0,5	0,06	2,2	3,1	29,1
	10	210	0,5	0,06	3,0	4,4	36,3
	11	170	0,1	0,06	2,2	3,2	29,6
	12	170	0,9	0,06	2,8	3,9	29,3
	13	170	0,5	0,04	3,7	5,3	36,3
	14	170	0,5	0,08	3,3	5,0	38,8
	15	170	0,5	0,06	1,6	2,4	23,3
	16	170	0,5	0,06	2,3	3,5	34,7
	17	170	0,5	0,06	2,3	3,4	31,8
	18	170	0,5	0,06	1,7	2,3	21,1
	19	170	0,5	0,06	3,3	5,2	45,2
	20	170	0,5	0,06	4,0	5,7	42,1


Shalom Eliahou

Eshita Mazumdar

Iterated sumsets and Hilbert functions

Keywords: Iterated sumsets and Hilbert functions Shalom Eliahou, Eshita Plünnecke's inequality, Standard Graded Algebra, Macaulay's Theorem, Binomial representation 11P70, 13P25, 05E40

whether they are published or not. The documents may come

Introduction

Let A be a nonempty finite subset of an abelian group (G, +). For any h ∈ N + = {1, 2, . . . }, we denote by hA the h-fold iterated sumset of A, i.e.

hA = A + • • • + A = {x 1 + • • • + x h | x i ∈ A for all 1 ≤ i ≤ h}.
As usual, we set 0A = {0}. A central problem in additive combinatorics is to understand the behavior of |hA| as h grows. Asymptotically, it is known that |hA| is eventually polynomial in h. See e.g. [START_REF] Khovanskii | Newton Polyhedron, Hilbert Polynomial, and Sums of Finite Sets[END_REF][START_REF] Khovanskii | Sums of Finite Sets, Orbits of Commutative Semigroups, and Hilbert Functions[END_REF][START_REF] Nathanson | Additive Number Theory, Inverse Problems and the Geometry of Sumsets[END_REF]. But not much is known about this polynomial and, for h small, the behavior of |hA| may wildly depend on the structure, or lack thereof, of A. For instance, if A is a subset of Z such that |A| = n, then

h(n -1) + 1 ≤ |hA| ≤ n -1 + h h ,
both bounds being attained in suitable cases: arithmetic progressions for the lower bound, and so-called B h -sets for the upper bound. The latter is best understood by noting that this binomial coefficient counts the number of monomials of degree h in |A| commuting variables. See e.g. [18, Sections 2.1 and 4.5] or [START_REF] Eliahou | Near-misses in Wilf's conjecture[END_REF]Section 3.2].

Here we address the following question. If h ≥ 2 and |hA| is known, what estimates on |(h -1)A| and |(h + 1)A| can one derive? The classical answer, given by Plünnecke's inequality and based on graph theory [START_REF] Peeva | Graded syzygies. Algebra and Applications[END_REF], is as follows:

|(h -1)A| ≥ |hA| (h-1)/h . (1)
See also [START_REF] Ruzsa | Sumsets and structure. Combinatorial number theory and additive group theory[END_REF][START_REF] Nathanson | Additive Number Theory, Inverse Problems and the Geometry of Sumsets[END_REF][START_REF] Tao | Additive combinatorics[END_REF]. In this paper, we derive this bound with a completely new approach, and we significantly improve it along the way. Our approach relies on Macaulay's classical 1927 theorem characterizing the Hilbert functions of standard graded algebras [START_REF] Macaulay | Some properties of enumeration in the theory of modular systems[END_REF]. We apply that theorem to a suitable standard graded K-algebra R = R(A) = ⊕ h≥0 R h having the property

dim K R h = |hA|
for all h ≥ 0. Using a recent condensed version of Macaulay's theorem [START_REF] Eliahou | Wilf's conjecture and Macaulay's theorem[END_REF], we improve (1) as follows. Denote

θ(x, h) = h x x h

1/h for x ∈ R and h ∈ N. If |A| ≥ 2, our improved bound implies

|(h -1)A| ≥ θ(x, h)|hA| (h-1)/h , (2)
where x is the unique real number larger than h such that |hA| = x h . This ensures θ(x, h) > 1. In fact, the factor θ(x, h) often exceeds 1.5 and even 2, as shown in Sections 5.2 and 5.3. For instance, for h = 12 we have θ(x, 12)) > 2.013 for all x ≥ 50. This implies in turn that if A satisfies |12A| ≥ 121,400,000,000, then

|11A| ≥ 2 |12A| 11/12 .
The wide occurrence of the case θ(x, h) ≥ 2 is described in more detail in Section 5.3. Remarkably, for x large enough and suitable values of h depending on x, the factor θ(x, h) approaches e ≈ 2.718, the basis of the natural logarithm. For instance, this occurs for all x ≥ 10 6 at h = 3000. See also Section 5.4, where strong evidence suggests that lim x→∞ θ(x, x 1/2 ) = e. Three general remarks are in order here.

Remark 1.1. Our results are stated for finite subsets of an abelian group G, but they hold more generally if G is a commutative semigroup, as in [START_REF] Nathanson | Growth of sumsets in abelian semigroups[END_REF] for instance.

Remark 1.2. Commutative algebra has already been applied to estimate the growth of iterated sumsets. In particular, the Hilbert polynomial of graded modules has been used to determine the asymptotic behavior of the function h → |hA|, and more generally of the function [START_REF] Khovanskii | Newton Polyhedron, Hilbert Polynomial, and Sums of Finite Sets[END_REF][START_REF] Khovanskii | Sums of Finite Sets, Orbits of Commutative Semigroups, and Hilbert Functions[END_REF][START_REF] Nathanson | Growth of sumsets in abelian semigroups[END_REF][START_REF] Nathanson | Additive Number Theory, Inverse Problems and the Geometry of Sumsets[END_REF]. However, to the best of our knowledge, the only previous application of Macaulay's theorem to additive combinatorics is in [START_REF] Eliahou | Wilf's conjecture and Macaulay's theorem[END_REF], where the above-mentioned condensed version is established and applied to yield an asymptotic solution of Wilf's conjecture on numerical semigroups. Remark 1.3. Another way of comparing |hA| with |(h-1)A| has been made, at least for A ⊂ Z, by seeking to bound the difference |hA| -|(h -1)A| from below rather than the quotient |hA|/|(h-1)A| from above [START_REF] Lev | Structure theorem for multiple addition and the Frobenius problem[END_REF]. In the study of the difference |hA| -|(h -1)A|, a main tool is Kneser's theorem, whereas for the quotient |hA|/|(h -1)A|, the classical one is Plünnecke's inequality, and an additional one is now Macaulay's theorem as made plain in this paper.

(h 1 , . . . , h r ) → |B + h 1 A 1 + • • • + h r A r |. See
There is a vast literature on Plünnecke's inequality, its rich applications to additive combinatorics and its successive refinements, such as the Plünnecke-Ruzsa inequality for instance [START_REF] Ruzsa | Arithmetical progressions and the number of sums[END_REF]. Besides dedicated chapters in [START_REF] Ruzsa | Sumsets and structure. Combinatorial number theory and additive group theory[END_REF][START_REF] Nathanson | Additive Number Theory, Inverse Problems and the Geometry of Sumsets[END_REF][START_REF] Tao | Additive combinatorics[END_REF], see also the nice survey [START_REF] Petridis | The Plünnecke-Ruzsa inequality: an overview[END_REF] and its many references.

The contents of this paper are as follows. In Section 2, we construct a graded algebra R(A) whose Hilbert function exactly models the sequence |hA| h≥0 . In Section 3, we recall Macaulay's theorem on Hilbert functions and the recent condensed version that we shall use. We prove our main results in Section 4. The first ones, Theorems 4.3 and 4.4, are obtained by applying Macaulay's theorem and its condensed version to the algebra R(A). The strength of these results is then illustrated with the specific case |5A| = 100. Here, Plünnecke's inequality implies |4A| ≥ 40 and |6A| ≤ 251, whereas our method yields much sharper and almost optimal bounds, namely |4A| ≥ 61 and |6A| ≤ 152. As our next main result, Theorem 4.9, we derive the Plünnecke-based estimate (1) from Theorem 4.4 and improve it by some multiplicative factor θ(x, h) > 1. The numerical behavior of that factor is studied in Section 5 and shown to often exceed 1.5 and even 2. In Section 6, we give a presentation of R(A) by generators and relations. We conclude the paper in Section 7 with related questions and remarks.

The graded algebra R(A)

Let A be a finite subset of an abelian group. Here we associate to A a standard graded algebra R(A) whose Hilbert function models the sequence |hA|. We start by recalling some basic terminology. Definition 2.1. A standard graded algebra is a commutative algebra R over a field K endowed with a vector space decomposition R = ⊕ i≥0 R i such that R 0 = K, R i R j ⊆ R i+j for all i, j ≥ 0, and which is generated as a K-algebra by finitely many elements in R 1 .

It follows from the definition that each R i is a finite-dimensional vector space over K. Moreover, as R is generated by R 1 , we have

R i R j = R i+j for all i, j ≥ 0, whence R i = R i 1 , the i-fold iterated productset of R 1 . Definition 2.2. Let R = ⊕ i≥0
R i be a standard graded algebra. The Hilbert function of R is the map i → d i associating to each i ∈ N the dimension

d i = dim K R i
of R i as a vector space over K.

Thus d 0 = 1, and R is generated as a K-algebra by any d 1 linearly independent elements of R 1 .

Let now (G, +) be an abelian group. Consider the group algebra K[G] of G. Its canonical K-basis is the set of symbols {t g | g ∈ G}, and its product is induced by the formula

t g 1 t g 2 = t g 1 +g 2 for all g 1 , g 2 ∈ G. Consider now S = K[G][Y ],
the one-variable polynomial algebra over K [G]. Then S has for K-basis the set

B = {t g Y n | g ∈ G, n ∈ N},
and the product of any two basis elements is given by

t g 1 Y n 1 • t g 2 Y n 2 = t g 1 +g 2 Y n 1 +n 2
for all g 1 , g 2 ∈ G and all n 1 , n 2 ∈ N. The degree of a basis element is defined as

deg(t g Y n ) = n
for all g ∈ G and all n ∈ N. This endows S with the structure of a graded algebra. Thus S = ⊕ h≥0 S h , where S h is the K-vector space with basis the set {t g Y h | g ∈ G}.

Definition 2.3. Let A = {a 1 , . . . , a n } be a nonempty finite subset of G. We define R(A) to be the K-subalgebra of S spanned by the set {t a 1 Y, . . . , t an Y }.

Thus R(A), being finitely generated over K by elements of degree 1, is a standard graded algebra. We then have R = ⊕ h≥0 R h , where R h is the K-vector space with basis the set

{t b Y h | b ∈ hA}. It follows that dim R h = |hA| (3)
for all h ≥ 0, as desired.

For future work on R(A), it is algebraically important to determine the relations between its given generators t a i Y . This is done in Section 6.

Macaulay's theorem

We now turn to Macaulay's theorem [START_REF] Macaulay | Some properties of enumeration in the theory of modular systems[END_REF] and a recent condensed version of it [START_REF] Eliahou | Wilf's conjecture and Macaulay's theorem[END_REF]. Macaulay's theorem gives a necessary and sufficient condition for a numerical function N → N to be the Hilbert function of some standard graded algebra. It rests on the so-called binomial representations of integers. Here is some background information.

Proposition 3.1. Let a ≥ i ≥ 1 be positive integers. There are unique integers a i > a i-1 > • • • > a 1 ≥ 0 such that a = i j=1 a j j .
Proof. See e.g. [START_REF] Bruns | Cambridge Studies in Advanced Mathematics[END_REF][START_REF] Peeva | Graded syzygies. Algebra and Applications[END_REF].

This expression is called the ith binomial representation of a. Producing it is computationally straightforward: take for a i the largest integer such that a i i ≤ a, and complete a i i by adding to it the (i -1)th binomial representation of a -a i i . We omit trails of 0's, if any. For instance, for a = 10 and i = 3, we abbreviate 10 = 5 3 + 1 2 + 0 1 as simply 10 = 5 3 .

Notation 3.2. Let a ≥ i ≥ 1 be positive integers. Let a = i j=1 a j j be its ith binomial representation. We denote a i = i j=1 a j + 1 j + 1 and 0 i = 0.

Note that the defining formula of a i yields the (i + 1)th binomial representation of the integer it sums to.

Here is one half of Macaulay's classical result, constraining the possible Hilbert functions of standard graded algebras [START_REF] Macaulay | Some properties of enumeration in the theory of modular systems[END_REF].

Theorem 3.3. Let R = ⊕ i≥0 R i be a standard graded algebra over a field K, with Hilbert function d i = dim K R i . Then for all i ≥ 1, we have d i+1 ≤ d i i . (4)
Remarkably, the converse also holds in Macaulay's theorem, but we shall not need it here. That is, satisfying (4) for all i ≥ 1 characterizes the Hilbert functions of standard graded algebras. See e.g. [START_REF] Bruns | Cambridge Studies in Advanced Mathematics[END_REF][START_REF] Mermin | Hilbert functions and lex ideals[END_REF][START_REF] Peeva | Graded syzygies. Algebra and Applications[END_REF]. Then m i+1 ≤ m i i for all i = 1, . . . , 5 as readily checked. Hence there exists a standard graded algebra R = ⊕ j≥0 R j whose values of dim R i for i = 0, . . . , 6 are exactly modeled by the sequence (m 0 , . . . , m 6 ). For instance, one may take R = S/J, where S = K[X 1 , . . . , X 5 ] and J = (X 3 5 , X 4 X 2 5 , X 3 3 X 2 5 ).

A condensed version

For our new derivation of the Plünnecke-based estimate (1), we shall need the following condensed version of Macaulay's theorem established in [START_REF] Eliahou | Wilf's conjecture and Macaulay's theorem[END_REF]. For m ∈ N and x ∈ R, denote as usual

x m = x(x -1) • • • (x -m + 1) m! = m-1 i=0 x -i m -i .
In particular, x 0 = 1. We shall constantly need the following observations. Lemma 3.5. Let i ≥ 1 be an integer. Then the map y → y i is an increasing continuous bijection (in fact, a homeomorphism) from

[i -1, ∞) to [0, ∞).
In particular, for any real numbers y 1 , y 2 ≥ i -1, we have

y 1 ≤ y 2 ⇐⇒ y 1 i ≤ y 2 i . (5)
Proof. A direct consequence of Rolle's theorem. See e.g. [START_REF] Eliahou | Wilf's conjecture and Macaulay's theorem[END_REF]Lemma 5.6].

Lemma 3.6. Let h, d ≥ 1 be positive integers. Then there exists a unique real number x ≥ h such that d = x h .

Proof. By the above lemma, there is a unique real number

x ≥ h -1 such that d = x h . Since d ≥ 1, we have x h ≥ h h . Hence x ≥ h by (5).
Here is the condensed version of Macaulay's theorem that we shall use in the next section. Theorem 3.7. Let R = ⊕ i≥0 R i be a standard graded algebra over the field K, with Hilbert function

d i = dim K R i for i ≥ 0. Let h ≥ 1 be an integer. Let x ≥ h -1 be the unique real number such that d h = x h . Then d h-1 ≥ x -1 h -1 and d h+1 ≤ x + 1 h + 1 .
Proof. See [START_REF] Eliahou | Wilf's conjecture and Macaulay's theorem[END_REF].

Main results

Let

|(h -1)A| ≥ |hA| (h-1)/h . (6)
Indeed, the general case is implied by [START_REF] Khovanskii | Newton Polyhedron, Hilbert Polynomial, and Sums of Finite Sets[END_REF], as shown by induction on h:

|iA| (6) ≥ (|(i + 1)A| i/(i+1) ind.hyp. ≥ (|hA| (i+1)/h ) i/(i+1) = |hA| i/h .
Consequently, in the sequel, we mainly focus on comparing |hA| with |(h -1)A| and/or |(h + 1)A|. In this spirit, a particular case of Plünnecke's Theorem 4.1 is the estimate

|(h + 1)A| ≤ |hA| (h+1)/h (7)
for all h ≥ 1.

In comparison, here is our first main result, obtained by applying Macaulay's Theorem 3.3 to the standard graded algebra R(A) defined in Section 2.

Theorem 4.3. Let A be a nonempty finite subset of an abelian group G. Let h ≥ 1 be an integer. Then

|(h + 1)A| ≤ |hA| h . ( 8
)
The strength of Theorem 4.3 is illustrated in Section 4.1, with a concrete example showing that (8) may be much sharper than [START_REF] Khovanskii | Sums of Finite Sets, Orbits of Commutative Semigroups, and Hilbert Functions[END_REF]. In fact, the improvement of the former over the latter is systematic, as shown by Theorem 4.9, Corollary 4.10 and Remark 4.11. See also a comment in Section 7.

Proof. Let R = R(A) be the standard graded algebra associated to A as defined in Section 2. We have R = ⊕ h≥0 R h , where R h denotes the homogeneous subspace of R of degree h. By (3) we have dim R h = |hA| for all h ≥ 0. Hence, for h ≥ 1, a direct application of Theorem 3.3 yields the claimed upper bound [START_REF] Lev | Structure theorem for multiple addition and the Frobenius problem[END_REF].

Let us now apply Theorem 3.7, the condensed version of Macaulay's theorem. We obtain the following more flexible bounds from which we shall derive and improve [START_REF] Khovanskii | Newton Polyhedron, Hilbert Polynomial, and Sums of Finite Sets[END_REF]. Then

|(h -1)A| ≥ x -1 h -1 and |(h + 1)A| ≤ x + 1 h + 1 .
Proof. As above, let R = R(A) be the standard graded algebra associated to A with its decomposition R = ⊕ h≥0 R h into the direct sum of its homogeneous subspaces of given degree, where dim R h = |hA| for all h ≥ 0. The claimed bounds follow from Theorem 3.7 applied to R(A).

Remark 4.5. While more handy, the upper bound in Theorem 4.4 is slightly weaker than in Theorem 4.3. Indeed, still with x ≥ h such that |hA| = x h , we have

|hA| h ≤ x + 1 h + 1 .
This follows from [3, Theorem 5.9].

Given |hA|, the lower bound on |(h -1)A| provided by Theorem 4.4 may be up to 2.71 times better, in suitable circumstances, than the one provided in (6) by Plünnecke's inequality. This will be shown in Sections 4.2 and 5. We start with a concrete example demonstrating the strength of Theorems 4.3 and 4.4.

An example: the case |5A| = 100

Let A be a subset of an abelian group such that |5A| = 100. The Plünneckebased bounds given by ( 6

Indeed, let x ≥ 5 be the unique real number such that x 5 = 100. Then 8.69 < x < 8.7, as follows from We conjecture that these bounds are optimal for sets of integers. As seen here, the improvement provided by Theorem 4.4 is already quite good. How good is it in general? We investigate this question in the sequel.

Macaulay vs Plünnecke

As our next main result, we show that Plünnecke's Theorem 4.1 also follows from our Macaulay-based Theorem 4.4, and we significantly strengthen it by a multiplicative factor which may exceed 2.71 in suitable circumstances. Notation 4.8. For a positive integer h and a real number x ≥ h, we set

θ(x, h) = h x x h 1/h
. Theorem 4.9. Let A be a nonempty finite subset of an abelian group G. Let h ≥ 2 be an integer. Then

|(h -1)A| ≥ θ(x, h) |hA| (h-1)/h ,
where x ∈ R satisfies x ≥ h and |hA| = x h .

Proof. Theorem 4.4 yields Proof. By Theorem 4.9, we only need to show θ(x, h) ≥ 1, or equivalently,

|(h -1)A| ≥ x -1 h -1 . ( 13
) Now x -1 h -1 = h x x h (14) since x h = h-1 i=0 x -i h -i = x h h-1 i=1 x -i h -i = x h x -1 h -1 . Hence |(h -1)A| h ≥ x -1 h -1 h = h x h x h h = h x h x h x h h-1 = h x h x h |hA| h-1 . Therefore |(h -1)A| h ≥ θ(x, h) h |hA| h-1 , as desired.
θ(x, h) h ≥ 1. Now θ(x, h) h = h x h x h = h-1 i=0 h(x -i) x(h -i) , (15)
and h(x

-i) ≥ x(h -i) for all 0 ≤ i ≤ h -1 since h ≤ x.
Remark 4.11. In fact, we have θ(x, h) > 1 whenever h ≥ 2 and |hA| ≥ 2. Inded, since |hA| = x h with x ≥ h, it follows that x > h, whence h(x -1) > x(h -1), implying in turn θ(x, h) h > 1 by [START_REF] Plünnecke | Eine zahlentheoretische Anwendung der Graphentheorie[END_REF].

We close this section with an equivalent formulation of Theorem 4.9. It provides a nice inequality between |(h -1)A| and |hA|, yet less suited to comparison purposes with Plünnecke's inequality. Theorem 4.12. Let A be a nonempty finite subset of an abelian group G. Let h ≥ 2 be an integer. Then

|(h -1)A| ≥ h x |hA|,
where x ∈ R satisfies x ≥ h and |hA| = x h .

Proof. Directly follows from Theorem 4.9 and the formulas

θ(x, h) = h x x h 1/h = h x |hA| (h-1)/h .
Alternatively, directly follows from Theorem 4.4 and formula [START_REF] Petridis | The Plünnecke-Ruzsa inequality: an overview[END_REF].

Corollary 4.13. Let A be a nonempty finite subset of an abelian group G. For all i ≥ 1, let x i ∈ R satisfy x i ≥ i and

x i i = |iA|. Let h ≥ 2 be an integer. For all 1 ≤ i ≤ h -1, we have |hA| ≤ h j=i+1 x j /j |iA|.
Proof. Straightforward consequence of the above theorem.

Behavior of θ(x, h)

We now study the numerical behavior of the function θ(x, h). Denote e ≈ 2.718, the basis of the natural logarithm. We show that 1 < θ(x, h) < e whenever x > h ≥ 2, and that θ(x, h) asymptotically tends to e in suitable circumstances. This section is slightly more informal in nature. Numerical computations and graphics were done with Mathematica 10 [20].

Proposition 5.1. For all h ∈ N, x ∈ R such that x > h ≥ 2, we have

1 < θ(x, h) < e.
Proof. The lower bound follows from [START_REF] Plünnecke | Eine zahlentheoretische Anwendung der Graphentheorie[END_REF] and Remark 4.11. As for the upper bound, we have

x h ≤ x h h! = x h h h h h h! < x h h h e h since h h h! < k∈N h k k! = e h . It follows that θ(x, h) = h x x h 1/h < h x x h e = e.
We shall also need to invoke the monotonicity of θ(x, h) in x.

Proposition 5.2. For a fixed integer h ≥ 2, the map

x → θ(x, h) from [h, ∞) to [1, ∞) is strictly increasing.
Proof. It is equivalent to show that the map x → θ(x, h) h is strictly increasing. This easily follows from the positivity of its derivative. Details are left to the reader.

Asymptotics

We provide here, somewhat informally, a good approximation of θ(x, h) together with its asymptotic behavior as x grows. Recall Stirling's approximation of n! for large n:

n! ∼ √ 2πn n e n .
On the other hand, the bounds below are valid for all n ≥ 1:

√ 2π n n+1/2 e -n ≤ n! ≤ e n n+1/2 e -n .
This yields the following well known approximation of n k for n much larger than k, see e.g. [19]:

n k ∼ (n/k -1/2) k e k √ 2πk .
As a consequence, here is the asymptotic behavior of θ(x, h) when x grows.

Proposition 5.3. Let h ≥ 2 be an integer. Then

θ(x, h) ∼ (1 -h/(2x)) e (2πh) 1/(2h) = (2x -h) e 2x(2πh) 1/(2h) .
In particular, lim

x→∞ θ(x, h) = (2πh) -1/(2h) e.
Proof. Directly follows from the above approximation of the binomial coefficients.

When θ(x, h) ≥ 1.5

Our multiplicative improvement factor θ(x, h) over the Plünnecke-based estimate

|(h -1)A| ≥ |hA| (h-1)/h (16)
exceeds 1.5 quite early in terms of x or h. Indeed, one observes that the smallest integer x for which θ(x, h) ≥ 1.5 for some integer h is x = 10, specifically at h = 4 and 5. Even starting at h = 3, we have θ(x, 3) ≥ 1.509 [START_REF] Ruzsa | Arithmetical progressions and the number of sums[END_REF] for all x ≥ 12. As an example of application, these observations, together with Theorem 4.9, yield the following improvements of ( 16) for h = 3, 4, 5.

Corollary 5.4. Let A be a finite subset of an abelian group G. Then

|3A| ≥ 220 =⇒ |2A| ≥ 3 2 |3A| 2/3 , |4A| ≥ 210 =⇒ |3A| ≥ 3 2 |4A| 3/4 , |5A| ≥ 252 =⇒ |4A| ≥ 3 2 |5A| 4/5 .
Proof. For h = 3, let x be the unique real number greater than 3 such that |3A| = x 3 . Since |3A| ≥ 220 = 12 3 , we have x ≥ 12, whence θ(x, 3) ≥ 1.5 by [START_REF] Ruzsa | Arithmetical progressions and the number of sums[END_REF]. The conclusion follows from Theorem 4.9. For h = 4 and 5, we have 10 4 = 210 and 10 5 = 252. The rest of the proof is similar, using the above-mentioned estimates θ(x, 4), θ(x, 5) ≥ 1.5 for all x ≥ 10.

When θ(x, h) ≥ 2

We now examine circumstances guaranteeing θ(x, h) ≥ 2, a case of interest where our bound in Theorem 4.9 is at least twice better than [START_REF] Ruzsa | Sumsets and structure. Combinatorial number theory and additive group theory[END_REF]. As it turns out, for x large, one has θ(x, h) ≥ 2 for almost all integers h between 6 and x/2 . We also describe cases where θ(x, h) gets very close to its upper bound e.

So, under what minimal circumstances, in terms of h or of x, do we have θ(x, h) ≥ 2? First note that if y ≥ h -1 then θ(y, h) < lim x→∞ θ(x, h), as follows from Proposition 5.2. Moreover, lim x→∞ θ(x, h 1 ) ≤ lim x→∞ θ(x, h 2 ) whenever h 1 ≤ h 2 , as follows from Proposition 5.3.

That being said, consider the case h = 5. Since lim x→∞ θ(x, 5) < 1.926 by Proposition 5.3, the values 1 ≤ h ≤ 5 are excluded for the occurrence of θ(x, h) ≥ 2. However, already h = 6 qualifies, as lim x→∞ θ(x, 6) > 2.007. More precisely, we have

θ(x, 6) ≥ 2 (18)
for all x ≥ 1210, the least integer with that property. If now h is allowed to grow, then θ(x, h) ≥ 2 may occur for much smaller values of x. Indeed, the smallest x ∈ N for which θ(x, h) ≥ 2 for some h is x = 48, namely at h = 11 and 12. More precisely, we have θ(48, 11) > 2.001, θ(48, 12) > 2.002, θ(48, 10) < 1.997, θ(48, 13) < 1.999.

See Figure 1.

In fact, when x goes to infinity, then θ(x, h) ≥ 2 holds for almost all positive integers h ≤ x/2. Indeed, as observed in [START_REF] Tao | Additive combinatorics[END_REF], we have θ(x, 6) ≥ 2 for all x ≥ x 0 = 1210. Now, numerical computations at x 0 yield

θ(x 0 , h) ≥ 2 ∀h ∈ [6, x 0 /2 -10] ∩ N. (19)
Together with Theorem 4.9, this yields the following factor 2 improvement over the Plünnecke-based estimate (16).

Corollary 5.5. Let h be an integer such that 6 ≤ h ≤ 595. Let A be a subset of an abelian group G such that |hA| ≥ 2h+20 h . Then As yet another instance, for x 1 = 10 6 now, one has an almost identical statement as in (19) for x 0 = 1210, namely

|(h -1)A| ≥ 2|hA| (h-1)/h .
θ(x 1 , h) ≥ 2 ∀h ∈ [6, x 1 /2 -19] ∩ N. (20)
Statements ( 19) and ( 20) are no accident, as hinted by the following result.

Proposition 5.6. One has lim x→∞ θ(x, x/2 ) = 2.

Proof. Using Stirling's approximation formula of n!, one readily sees that

θ(n, n/2 ) ≈ 2 2 πn 1/n
, which proves the claim since lim n→∞ (cn) -1/n = 1 for any constant c > 0.

The highest point

For fixed x, the general shape of θ(x, h) when h runs from 1 to x is well illustrated by Figure 1 for x = 48. It would be desirable to determine the highest point of that curve, and in particular the integer 1 ≤ h ≤ x maximizing θ(x, h). We do not have yet a precise answer. Nevertheless, by computing derivatives of the approximation of θ(x, h) provided by Proposition 5.3, one sees that for fixed x, ∂ ∂h 2x -h (2πh) 1/(2h) > 0 ⇐⇒ 2h 2 < (2x -h)(ln(2πh) -1).

(21)

Thus, for x fixed, the sought-for integer h maximizing θ(x, h) occurs when

2h 2 ≈ (2x -h)(ln(2πh) -1). (22)
For instance, for x 0 = 100, the maximum of θ(x 0 , h) is reached at h = 18, for which θ(100, 18) ≈ 2.177. Hence θ(x, 18) ≥ 2.177 for all x ≥ 100, as follows from Proposition 5.2.

For h fixed

In the opposite direction, for h fixed, it is easy to locate the real number x 1 ≥ h maximizing θ(x, h). Indeed, using (21), we find

x 1 ≈ 1 2 2h 2 ln(2πh) -1 + h .
This suggests that

lim x→∞ θ(x, x 1/2 ) = e,
as is fully confirmed by numerical experiments. As a concrete illustration, here are instances where θ(x, h) gets very close to e:

• For all x ≥ 200000 and all 1200 ≤ h ≤ 1300, one has θ(x, h) ≥ 2.70.

• Similarly, for all x ≥ 1100000 and all 2600 ≤ h ≤ 3700, one has θ(x, h) ≥ 2.71.

A presentation of R(A)

Reusing the notation of Section 2, let A = {a 1 , . . . , a n } be a nonempty finite subset of an abelian group (G, +). For future use, it is algebraically necessary to determine the relations between the given generators t a i Y of the associated algebra R(A). Our aim here is thus to identify R(A) as the quotient of the polynomial algebra K[X 1 , . . . , X n ] by a suitable homogeneous ideal I.

Notation 6.1. For α = (α 1 , . . . , α n ) ∈ N n , let X α = X α 1 1 • • • X αn n denote the corresponding monomial in K[X 1 , . . . , X n ].
We denote the set of these monomials by M = {X α | α ∈ N n } .

Let ϕ : K[X 1 , . . . , X n ] → R(A) be the surjective morphism induced by ϕ(X i ) = t a i Y for all i. On the set M , we define the equivalence relation

u ∼ v ⇐⇒ ϕ(u) = ϕ(v) for all u, v ∈ M . Equivalently, let us write u = X α , v = X β with α = (α 1 , . . . , α n ), β = (β 1 , . . . , β n ) ∈ N n . Then X α ∼ X β ⇐⇒ i α i = i β i , i α i a i = i β i a i .
In particular, equivalent monomials have the same degree, where as usual deg(X α ) = i α i . We shall need the notion of simple polynomial relative to ∼.

Definition 6.2. Let f ∈ K[X 1 , . . . , X n ].
We say that f is simple if f = 0 and all monomials occurring in f are equivalent under ∼.

Observe that a simple polynomial is homogeneous. Indeed, equivalent monomials under ∼ have the same degree as observed above. Moreover, every nonzero polynomial g ∈ K[X 1 , . . . , X n ] may be decomposed, in a unique way up to order, as the sum g = f 1 + • • • + f r of maximal simple polynomials f i , in the sense that for all i = j, the monomials occurring in f i are nonequivalent under ∼ to those of f j . The f i are obtained by simply regrouping the monomials of f into maximal equivalence classes. We shall refer to the f i as the simple components of f . See e.g. [2, p. 232] and [5, p. 346], where similar notions were used. Lemma 6.3. Let g ∈ ker(ϕ)\{0}. Then every simple component of g belongs to ker(ϕ).

Proof. Let f be a simple component of g. We must show ϕ(f ) = 0. Since f is simple, it is homogeneous of some degree h. Write f = i λ i u i , where λ i ∈ K \ {0} for all i and where the u i are pairwise distinct monomials. Since the u i are pairwise equivalent under ∼, we have ϕ(

u i ) = t b Y h for some b ∈ hA independent of i. Hence ϕ(f ) = ( i λ i )t b Y h .
Now, for any monomial v occurring in g but not in f , we have ϕ(v) = t b Y h as v is non-equivalent to the u i . Since ϕ(g) = 0, it follows that i λ i = 0. Hence ϕ(f ) = 0, as desired. Proposition 6.4. Let I ⊂ K[X 1 , . . . , X n ] be the ideal generated by the set {u -v | u, v ∈ M, u ∼ v}. Then ker(ϕ) = I.

Proof. We have I ⊂ ker(ϕ) by construction. Conversely, let 0 = f ∈ ker(ϕ). By Lemma 6.3, we may further assume that f is simple. Write f = r i=1 λ i u i , where λ i ∈ K \{0} for all i and where the u i are pairwise distinct monomials. Since ϕ(f ) = 0 and ϕ(u i ) = ϕ(u j ) for all i = j, it follows that r i=1 λ i = 0. Therefore λ r = -r-1 i=1 λ i , and so

f = r-1 i=1 λ i (u i -u r ).
Since u i ∼ u r for all i, it follows that u i -u r ∈ I. Hence f ∈ I, as desired.

Corollary 6.5. We have R(A) K[X 1 , . . . , X n ]/I.

Proof. By Noether's isomorphism theorem.

Concluding comments

We end this paper with a few related questions and remarks.

A first natural question is, how far from optimal are our new bounds? More precisely, let (G, +) be an abelian group, and let h, i, m be positive integers such that m ≤ |G|. Among all subsets A ⊆ G such that |hA| = m, what is

• (inverse problem) the best possible lower bound on |iA| for i ≤ h?

• (direct problem) the best possible upper bound on |iA| for i ≥ h? Accordingly, let us denote

ω G (h, i, m) = min A⊆G |iA| if i ≤ h, max A⊆G |iA| if i ≥ h,
where in both cases, the extremum is taken over all subsets A of G satisfying |hA| = m. Focusing here on the direct problem with i = h + 1, how large can ω G (h, h + 1, m) be? The upper bounds given successively by Plünnecke's inequality [START_REF] Khovanskii | Sums of Finite Sets, Orbits of Commutative Semigroups, and Hilbert Functions[END_REF] The last one is probably close to optimal. Indeed, for G = Z, we gave an example with |5A| = 100 and |6A| = 145, yielding ω Z (5, 6, 100) ≥ 145. Conjecture 4.7 implies that this is best possible, i.e. that ω Z (5, 6, 100) = 145.

ω G (h, h + 1, m) ≤ x + 1 h + 1 = x + 1 h + 1 m, (24)
ω G (h, h + 1, m) ≤ m h , (25)
As another natural question, can one specialize Macaulay's theorem by characterizing the Hilbert functions of all algebras of the form R(A) for finite subsets A of a given abelian group G? A positive answer would help tackle the former question.

Finally, in a sequel to this paper, we will show two more aspects of the strength of Theorem 4.3. The proof methods are quite different from the present ones, except that Macaulay's theorem remains central. First, we will show that Theorem 4.3 is asymptotically optimal : the upper bound it provides, namely |(h + 1)A| ≤ |hA| h for all h ≥ 1, is in fact an equality for h large enough. Second, we will show that Theorem 4.3 is best possible in the sense that, given any sequence of positive integers (d i ) i≥0 such that d 0 = 1 and

1 ≤ d i+1 ≤ d i i
for all i ≥ 1, there exists a finite subset A of a commutative semigroup (G, +) such that d h = |hA| for all h ≥ 0.

Together, the present paper and its forthcoming sequel raise the prospect that Macaulay's theorem, an almost century-old classical result from commutative algebra, may emerge as a powerful new tool in additive combinatorics.

Example 3 . 4 .

 34 Consider the sequence (m 0 , m 1 , m 2 , m 3 , m 4 , m 5 , m 6 ) = (1, 5, 15, 33, 61, 100, 152).

Theorem 4 . 4 .

 44 Let A be a nonempty finite subset of an abelian group G. Let h ≥ 2 be an integer and x ≥ h the unique real number such that |hA| = x h .

  ), namely |4A| ≥ 100 4/5 and |6A| ≤ 100 6/5 , yield |4A| ≥ 40, |6A| ≤ 251. In comparison, based on the condensed version of Macaulay's theorem, Theorem 4.4 yields the much sharper bounds |4A| ≥ 58, |6A| ≤ 161.

Conjecture 4 . 7 .

 47 Let A ⊂ Z be any subset satisfying |5A| = 100. Then |4A| ≥ 63 and |6A| ≤ 145.

Corollary 4 . 10 .

 410 Theorem 4.4 implies Plünnecke's Theorem 4.1.

Figure 1 :

 1 Figure 1: Values of θ(48, h) for h = 1, . . . , 48

Figure 2

 2 displays the case x = 1000.

Figure 2 :

 2 Figure 2: Values of θ(1000, h) for h = 1, . . . , 1000

  , Theorem 4.4 based on the condensed version of Macaulay's theorem, and Theorem 4.3 based on Macaulay's theorem proper, areω G (h, h + 1, m) ≤ m (h+1)/h ,(23)

  respectively, where x ≥ h satisfies x h = m. Applied to the case (h, m) = (5, 100) in Section 4.1, these bounds yield successively ω G (5, 6, 100) ≤    251 by (23), 161 by (24), 152 by (25).

  Proposition 4.6. Let A be a subset of an abelian group such that |5A| = 100.

	The inequality |(h + 1)A| ≤ |hA| h of Theorem 4.3 then yields (10). More
	precisely, we have
		|4A| ≥	7 4	+		6 3	+		4 2	= 61,	(11)
		|6A| ≤	9 6	+		8 5	+		5 4	+	4 3	+	3 2	= 152.	(12)
	Thus inequality (12) directly follows from Theorem 4.3. As for (11), if we had
	|4A| ≤ 60 =	7 4	+	6 3	+	3 2	+	2 1	, then that same theorem would imply
	|5A| ≤	8 5	+	7 4	+	4 3	+	3 2	= 100 -2, contrary to the hypothesis.
									8.69 5		≈ 99.42 and 8.7 5	≈ 100.2. Hence
					|4A| ≥		x -1 4	>	7.69 4	≈ 57.2,
					|6A| ≤		x + 1 6	<	9.7 6	≈ 161.99.
	Then
						|4A| ≥ 61, |6A| ≤ 152.	(10)
	Proof. The 5th binomial representation of 100 is given by
				100 =	8 5	+	7 4	+	4 3	+	3 2	+	2 1	.

This proves

[START_REF] Macaulay | Some properties of enumeration in the theory of modular systems[END_REF]

.

Theorem 4.3, based on the full version of Macaulay's theorem, yields even better bounds. Is Proposition 4.6 best possible for sets satisfying |5A| = 100? Strong evidence shows that it is not far from it. For instance, let A = {0, 1, 5, 8, 49} ⊂ Z. Then |5A| = 100 as required, and |4A| = 63, |6A| = 145.

Acknowledgments. This research was supported in part by the International Centre for Theoretical Sciences (ICTS) during a visit for the program -Workshop on Additive Combinatorics (Code: ICTS/wac2020/02). We are grateful to David Grynkiewicz for very useful discussions concerning this work during the ICTS Workshop.