[ { "id": "https://authors.library.caltech.edu/records/srsrf-x1974", "eprint_id": 113804, "eprint_status": "archive", "datestamp": "2023-08-22 13:16:29", "lastmod": "2023-10-23 23:14:32", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Chipeniuk-Karsten-O", "name": { "family": "Chipeniuk", "given": "Karsten O." } }, { "id": "Katz-N-H", "name": { "family": "Katz", "given": "Nets Hawk" } }, { "id": "Walker-Todd-B", "name": { "family": "Walker", "given": "Todd B." } } ] }, "title": "Households, auctioneers, and aggregation", "ispublished": "pub", "full_text_status": "public", "keywords": "Aggregation; Heterogeneous agents; Incomplete markets; Economics and Econometrics; Finance", "note": "\u00a9 2021 The Authors. Published by Elsevier. This is an open access article under the CC BY-NC-ND license. \n\nReceived 12 April 2021, Revised 25 October 2021, Accepted 13 November 2021, Available online 11 December 2021, Version of Record 15 December 2021. \n\nKatz acknowledges support from National Science Foundation, United States grant DMS 1266104. We would like to thank participants at the NBER Summer Institute, the Federal Reserve Banks of Chicago, Cleveland, Dallas and Richmond, Reserve Bank of New Zealand, Australian National University, University of Virginia, Bundesbank Spring Conference, Stanford Institute for Theoretical Economics, Konstanz Seminar on Monetary Theory and Policy, CEMLA Research Seminar and World Congress of the Econometric Society; and Tom Winberry, Kurt Mitman, Tony Smith, Sevin Yeltekin, and Eric Young for helpful comments. We thank the editor, Florin Bilbiie, for several excellent suggestions.\n\n
Published - 1-s2.0-S0014292121002713-main.pdf
Supplemental Material - 1-s2.0-S0014292121002713-mmc1.pdf
", "abstract": "We examine aggregation in the neoclassical growth model with aggregate shocks and uninsurable employment risk, as well as related environments. We introduce a Walrasian auctioneer whose job is to report to households all possible state-contingent future prices. Households take these as given when forming expectations and making optimal consumption/savings decisions, and the auctioneer adjusts her forecasts until markets clear. This natural dichotomy between the households and the auctioneer allows us to study each problem in isolation as well as to discuss the intersection. On the household side, we separate an explicit expression for the linear permanent income component of savings from a well-behaved nonlinear adjustment arising from precautionary behavior and incomplete markets. Equipped with this decomposition, we then study how economies aggregate in the presence of various auctioneer types that are popular in the literature. The steady-state auctioneer of Huggett (1997) and Aiyagari (1994) offers a paper-and-pencil analysis of aggregation that provides a bound on more complex environments. We provide an economic interpretation of the regression coefficients and explain the lack of time variation in the auctioneer of Krusell and Smith (1998). We also introduce a new numerical method which uses the empirical distribution of auctioneer forecasts to substantially improve solution accuracy in cases where the standard coefficient of determination and other well-known statistics prove to be misleading.", "date": "2022-01", "date_type": "published", "publication": "European Economic Review", "volume": "141", "publisher": "Elsevier", "pagerange": "Art. No. 103997", "id_number": "CaltechAUTHORS:20220308-454082000", "issn": "0014-2921", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220308-454082000", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1266104" } ] }, "doi": "10.1016/j.euroecorev.2021.103997", "primary_object": { "basename": "1-s2.0-S0014292121002713-main.pdf", "url": "https://authors.library.caltech.edu/records/srsrf-x1974/files/1-s2.0-S0014292121002713-main.pdf" }, "related_objects": [ { "basename": "1-s2.0-S0014292121002713-mmc1.pdf", "url": "https://authors.library.caltech.edu/records/srsrf-x1974/files/1-s2.0-S0014292121002713-mmc1.pdf" } ], "resource_type": "article", "pub_year": "2022", "author_list": "Chipeniuk, Karsten O.; Katz, Nets Hawk; et el." }, { "id": "https://authors.library.caltech.edu/records/7r4hq-qs353", "eprint_id": 113133, "eprint_status": "archive", "datestamp": "2023-08-22 10:25:09", "lastmod": "2023-10-23 22:56:55", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Guth-Larry", "name": { "family": "Guth", "given": "Larry" } }, { "id": "Katz-N-H", "name": { "family": "Katz", "given": "Nets Hawk" } }, { "id": "Zahl-Joshua", "name": { "family": "Zahl", "given": "Joshua" } } ] }, "title": "On the Discretized Sum-Product Problem", "ispublished": "pub", "full_text_status": "restricted", "keywords": "General Mathematics", "note": "\u00a9 The Author(s) 2020. Published by Oxford University Press. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model). \n\nReceived: 24 April 2018; Revision received: 31 October 2019; Accepted: 28 November 2019; Published: 13 January 2020. \n\nThe authors would like to thank Brendan Murphy, Victor Lie, and Jianan Li for comments and corrections to a previous draft of this manuscript. The authors would also like to thank the anonymous referees for corrections and suggestions. \n\nThis work was supported by a Simons Investigator Award [to L.G.]; and a NSERC Discovery Grant [to J.Z.].", "abstract": "We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if A \u2282 R is a (\u03b4,1/2)\u2081-set in the sense of Katz and Tao, then either A+A or A.A must have measure at least |A|1\u22121/68\u2060.", "date": "2021-07", "date_type": "published", "publication": "International Mathematics Research Notices", "volume": "2021", "number": "13", "publisher": "Oxford University Press", "pagerange": "9769-9785", "id_number": "CaltechAUTHORS:20220127-965114100", "issn": "1073-7928", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220127-965114100", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Simons Foundation" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)" } ] }, "doi": "10.1093/imrn/rnz360", "resource_type": "article", "pub_year": "2021", "author_list": "Guth, Larry; Katz, Nets Hawk; et el." }, { "id": "https://authors.library.caltech.edu/records/nezma-weh48", "eprint_id": 108042, "eprint_status": "archive", "datestamp": "2023-08-19 22:59:59", "lastmod": "2023-10-23 16:25:44", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Katz-N-H", "name": { "family": "Katz", "given": "Nets Hawk" } }, { "id": "Zahl-Joshua", "name": { "family": "Zahl", "given": "Joshua" } } ] }, "title": "A Kakeya maximal function estimate in four dimensions using planebrushes", "ispublished": "pub", "full_text_status": "public", "keywords": "Kakeya problem", "note": "\u00a9 2021 EMS Publishing House.\n\nSupported by NSF grant DMS 1565904.\n\nSupported by an NSERC Discovery grant. \n\nThe authors would like to thank Keith Rogers and the anonymous referees for comments and corrections on an earlier version of this manuscript.\n\nSubmitted - 1902.00989.pdf
", "abstract": "We obtain an improved Kakeya maximal function estimate in R\u2074 using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff's hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a plane. When Besicovitch sets do not have this property, the existing trilinear estimates of Guth\u2013Zahl can be used to bound the size of a Besicovitch set. In particular, we establish a maximal function estimate in R\u2074 at dimension 3.059. As a consequence, every Besicovitch set in R\u2074 must have Hausdorff dimension at least 3.059.", "date": "2020-08-20", "date_type": "published", "publication": "Revista Matem\u00e1tica Iberoamericana", "volume": "37", "number": "1", "publisher": "European Mathematical Society", "pagerange": "317-359", "id_number": "CaltechAUTHORS:20210212-133815036", "issn": "0213-2230", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210212-133815036", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1565904" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)" } ] }, "doi": "10.4171/rmi/1219", "primary_object": { "basename": "1902.00989.pdf", "url": "https://authors.library.caltech.edu/records/nezma-weh48/files/1902.00989.pdf" }, "resource_type": "article", "pub_year": "2020", "author_list": "Katz, Nets Hawk and Zahl, Joshua" }, { "id": "https://authors.library.caltech.edu/records/q15xy-ehs45", "eprint_id": 90740, "eprint_status": "archive", "datestamp": "2023-08-19 13:34:30", "lastmod": "2023-10-19 14:49:04", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Katz-N-H", "name": { "family": "Katz", "given": "Nets Hawk" } }, { "id": "Zahl-J", "name": { "family": "Zahl", "given": "Joshua" } } ] }, "title": "An improved bound on the Hausdorff dimension of Besicovitch sets in \u211d^3", "ispublished": "pub", "full_text_status": "public", "note": "\u00a9 2018 American Mathematical Society. \n\nReceived by the editors May 20, 2017, and, in revised form, September 16, 2017, and May 21, 2018. Published electronically: August 29, 2018. \n\nThe first author was supported by NSF grants DMS 1266104 and DMS 1565904. The second author was supported by an NSERC Discovery grant. \n\nThe authors would like to thank Terry Tao, Daniel Di Benedetto, and the anonymous referee for helpful comments and suggestions to an earlier version of this manuscript.\n\nSubmitted - 1704.07210.pdf
", "abstract": "We prove that every Besicovitch set in \u211d^3 must have Hausdorff dimension at least 5/2 + \u03f5_0 for some small constant \u03f5_0 > 0. This follows from a more general result about the volume of unions of tubes that satisfies the Wolff axioms. Our proof grapples with a new \"almost counterexample\" to the Kakeya conjecture, which we call the SL_2 example; this object resembles a Besicovitch set that has Minkowski dimension 3 but Hausdorff dimension 5/2. We believe this example may be an interesting object for future study.", "date": "2019-01", "date_type": "published", "publication": "Journal of the American Mathematical Society", "volume": "32", "number": "1", "publisher": "American Mathematical Society", "pagerange": "195-259", "id_number": "CaltechAUTHORS:20181108-083518614", "issn": "0894-0347", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181108-083518614", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1266104" }, { "agency": "NSF", "grant_number": "DMS-1565904" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)" } ] }, "doi": "10.1090/jams/907", "primary_object": { "basename": "1704.07210.pdf", "url": "https://authors.library.caltech.edu/records/q15xy-ehs45/files/1704.07210.pdf" }, "resource_type": "article", "pub_year": "2019", "author_list": "Katz, Nets Hawk and Zahl, Joshua" }, { "id": "https://authors.library.caltech.edu/records/qzf98-gbq57", "eprint_id": 89642, "eprint_status": "archive", "datestamp": "2023-08-19 12:56:39", "lastmod": "2023-10-18 22:53:33", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Katz-N-H", "name": { "family": "Katz", "given": "Nets Hawk" } }, { "id": "Rogers-Keith-M", "name": { "family": "Rogers", "given": "Keith M." } } ] }, "title": "On the polynomial Wolff axioms", "ispublished": "pub", "full_text_status": "public", "note": "\u00a9 Springer Nature Switzerland AG 2018. \n\nReceived: March 8, 2018; Revised: June 6, 2018; Accepted: July 20, 2018; First Online: 14 September 2018. \n\nThe first author would like to thank Josh Zahl for helpful discussions. In particular the proof of Lemma 2.2 came from a conversation with him. The second author would like to thank Jonathan Hickman for helpful discussions regarding the application to restriction. \n\nSupported by NSF grant DMS 1565904 and by MINECO Grants SEV-2015-0554 and MTM2017- 85934-C3-1-P.\n\nAccepted Version - 1802.09094
", "abstract": "We confirm a conjecture of Guth concerning the maximal number of \u03b4-tubes, with \u03b4-separated directions, contained in the \u03b4-neighborhood of a real algebraic variety. Modulo a factor of \u03b4^(\u2212\u03b5), we also prove Guth and Zahl's generalized version for semialgebraic sets. Although the applications are to be found in harmonic analysis, the proof will employ deep results from algebraic and differential geometry, including Tarski's projection theorem and Gromov's algebraic lemma.", "date": "2018-12", "date_type": "published", "publication": "Geometric and Functional Analysis", "volume": "28", "number": "6", "publisher": "Springer", "pagerange": "1706-1716", "id_number": "CaltechAUTHORS:20180914-100924111", "issn": "1016-443X", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180914-100924111", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1565904" }, { "agency": "Ministerio de Econom\u00eda, Industria y Competitividad (MINECO)", "grant_number": "SEV-2015-0554" }, { "agency": "Ministerio de Econom\u00eda, Industria y Competitividad (MINECO)", "grant_number": "MTM2017- 85934-C3-1-P" } ] }, "doi": "10.1007/s00039-018-0466-7", "primary_object": { "basename": "1802.09094", "url": "https://authors.library.caltech.edu/records/qzf98-gbq57/files/1802.09094" }, "resource_type": "article", "pub_year": "2018", "author_list": "Katz, Nets Hawk and Rogers, Keith M." }, { "id": "https://authors.library.caltech.edu/records/wfkbj-czf19", "eprint_id": 57015, "eprint_status": "archive", "datestamp": "2023-08-20 04:26:27", "lastmod": "2023-10-23 16:57:28", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Guth-L", "name": { "family": "Guth", "given": "Larry" } }, { "id": "Katz-N-H", "name": { "family": "Katz", "given": "Nets Hawk" } } ] }, "title": "On the Erd\u0151s distinct distances problem in the plane", "ispublished": "pub", "full_text_status": "public", "keywords": "distinct distances, Incidence geometry, polynomial ham sandwich, polynomial method, ruled surface", "note": "\u00a9 2015 Department of Mathematics, Princeton University. Received: 18 November 2010. Revised: 15 July 2014. Accepted: 14 April 2014.\n\nThe first author is partially supported by NSERC, by NSF grant DMS-0635607, and by the Monell Foundation. The second author is partially supported by NSF grant DMS-1001607. He would like to thank Michael Larsen for some very helpful discussions about algebraic geometry. He would also like to thank the Institute of Advanced Study for the use of its magnificent duck pond during a visit that resulted in this paper. Both authors would like to thank the helpful referee because of whom the exposition in the paper is significantly improved.\n\nPublished - annals-v181-n1-p02-p.pdf
", "abstract": "In this paper, we prove that a set of N points in R^2 has at least c^N_(logN) distinct distances, thus obtaining the sharp exponent in a problem of Erd\u0151s. We follow the setup of Elekes and Sharir which, in the spirit of the Erlangen program, allows us to study the problem in the group of rigid motions of the plane. This converts the problem to one of point-line incidences in space. We introduce two new ideas in our proof. In order to control points where many lines are incident, we create a cell decomposition using the polynomial ham sandwich theorem. This creates a dichotomy: either most of the points are in the interiors of the cells, in which case we immediately get sharp results or, alternatively, the points lie on the walls of the cells, in which case they are in the zero-set of a polynomial of suprisingly low degree, and we may apply the algebraic method. In order to control points incident to only two lines, we use the flecnode polynomial of the Rev. George Salmon to conclude that most of the lines lie on a ruled surface. Then we use the geometry of ruled surfaces to complete the proof.", "date": "2015-01", "date_type": "published", "publication": "Annals of Mathematics", "volume": "181", "number": "1", "publisher": "Princeton University, Department of Mathematics", "pagerange": "155-190", "id_number": "CaltechAUTHORS:20150427-131107573", "issn": "0003-486X", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150427-131107573", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)" }, { "agency": "NSF", "grant_number": "DMS-0635607" }, { "agency": "Monell Foundation" }, { "agency": "NSF", "grant_number": "DMS-1001607" } ] }, "doi": "10.4007/annals.2015.181.1.2", "primary_object": { "basename": "annals-v181-n1-p02-p.pdf", "url": "https://authors.library.caltech.edu/records/wfkbj-czf19/files/annals-v181-n1-p02-p.pdf" }, "resource_type": "article", "pub_year": "2015", "author_list": "Guth, Larry and Katz, Nets Hawk" }, { "id": "https://authors.library.caltech.edu/records/fmcwj-k5d18", "eprint_id": 62450, "eprint_status": "archive", "datestamp": "2023-08-20 04:11:48", "lastmod": "2023-10-25 17:10:01", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Katz-N-H", "name": { "family": "Katz", "given": "Nets" } }, { "id": "Tapay-A", "name": { "family": "Tapay", "given": "Andrew" } } ] }, "title": "A model for studying double exponential growth in the two-dimensional Euler equations", "ispublished": "pub", "full_text_status": "public", "keywords": "fluid mechanics, Euler equations, two-dimensional Euler equations", "note": "\u00a9 2015 Mathematical Sciences Publishers. \n\nReceived 16 Oct 2014. Revised 8 May 2015. Accepted 24 Jun 2015. \n\nBoth authors were partially supported by NSF grant DMS 1266104.\n\nPublished - apde-v8-n7-p04-s.pdf
Submitted - 1403.6867v1.pdf
", "abstract": "We introduce a model for the two-dimensional Euler equations which is designed to study whether or not double exponential growth can be achieved for a short time at an interior point of the flow.", "date": "2015", "date_type": "published", "publication": "Analysis & PDE", "volume": "8", "number": "7", "publisher": "Mathematical Sciences Publishers", "pagerange": "1675-1693", "id_number": "CaltechAUTHORS:20151130-132253511", "issn": "2157-5045", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20151130-132253511", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1266104" } ] }, "doi": "10.2140/apde.2015.8.1675", "primary_object": { "basename": "1403.6867v1.pdf", "url": "https://authors.library.caltech.edu/records/fmcwj-k5d18/files/1403.6867v1.pdf" }, "related_objects": [ { "basename": "apde-v8-n7-p04-s.pdf", "url": "https://authors.library.caltech.edu/records/fmcwj-k5d18/files/apde-v8-n7-p04-s.pdf" } ], "resource_type": "article", "pub_year": "2015", "author_list": "Katz, Nets and Tapay, Andrew" } ]