Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-25T11:14:25.867Z Has data issue: false hasContentIssue false
  • English
  • Français

Dichotomy and infinite combinatorics: the theorems of Steinhaus and Ostrowski

Published online by Cambridge University Press:  08 October 2010

N. H. BINGHAM  and
A. J. OSTASZEWSKI
N. H. BINGHAM
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ. e-mail: n.bingham@ic.ac.uk
A. J. OSTASZEWSKI
Affiliation:
Department of Mathematics, London School of Economics, London WC2A 2AE. e-mail: a.j.ostaszewski@lse.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We define combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the density topology on the line. Likewise we unify the subgroup theorem by reference to a Ramsey property. A combinatorial form of Ostrowski's theorem (that a bounded additive function is linear) permits the deduction of both the measure and category automatic continuity theorems for additive functions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 1 , January 2011 , pp. 1 - 22
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aarts, J. M. and Lutzer, D. J.Completeness properties designed for recognizing Baire spaces. Dissertationes Math. (Rozprawy Mat.), 116 (1974), 48.Google Scholar
[2]Arhangel'skii, A. V. Paracompactness and metrization. The method of covers in the classification of spaces, in General Topology III (ed. Arhangel'skii, A.V.), 1–70, Encyclopaedia Math. Sci., 51 (Springer, 1995).Google Scholar
[3]Banach, S. Théorie des opérations linéaires. Reprinted in Collected Works vol. II, 401–411, (PWN, Warszawa, 1979) (1st. edition 1932).Google Scholar
[4]Beiglböck, M. An ultrafilter approach to Jin's Theorem. Israel J. Math., to appear.Google Scholar
[5]Bingham, N. H. and Goldie, C. M.Extensions of regular variation, I: Uniformity and quantifiers. Proc. London Math. Soc. (3) 44 (1982), 473–496.CrossRefGoogle Scholar
[6]Bingham, N. H., Goldie, C. M. and Teugels, J. L. Regular variation. Encycl. Math. Appl. 27 (Cambridge University Press, 1987).Google Scholar
[7]Bingham, N. H. and Ostaszewski, A. J.Generic subadditive functions. Proc. Amer. Math. Soc. 136 (2008), 4257–4266.Google Scholar
[8]Bingham, N. H. and Ostaszewski, A. J.Infinite combinatorics and the foundations of regular variation. J. Math. Anal. Appl. 360 (2009), 518–529.CrossRefGoogle Scholar
[9]Bingham, N. H. and Ostaszewski, A. J.Very slowly varying functions – II. Colloq. Math. 116 (2009), 105–117.CrossRefGoogle Scholar
[10]Bingham, N. H. and Ostaszewski, A. J.Beyond Lebesgue and Baire: generic regular variation. Colloq. Math. 116 (2009), 119–138.CrossRefGoogle Scholar
[11]Bingham, N. H. and Ostaszewski, A. J.The index theorem of topological regular variation and its applications. J. Math. Anal. Appl. 358 (2009), 238–248.Google Scholar
[12]Bingham, N. H. and Ostaszewski, A. J.Automatic continuity: subadditivity, convexity, uniformity. Aequationes Math. 78 (2009), 257–270.CrossRefGoogle Scholar
[13]Bingham, N. H. and Ostaszewski, A. J.Infinite combinatorics in function spaces: category methods. Publ. Inst. Math. Béograd, 86 (100) (2009), 55–73.CrossRefGoogle Scholar
[14]Bingham, N. H. and Ostaszewski, A. J.Automatic continuity via analytic thinning. Proc. Amer. Math. Soc., 138 (2010), 907–919.Google Scholar
[15]Bingham, N. H. and Ostaszewski, A. J.Regular variation without limits. J. Math. Anal. Appl. 370 (2010), 322–338.CrossRefGoogle Scholar
[16]Bingham, N. H. and Ostaszewski, A. J. Kingman, category and combinatorics. Probability and mathematical genetics (Kingman Festschrift, J. F. C., ed. Bingham, N. H. and Goldie, C. M.), LMS Lecture Notes Series 378 (2010), 135–168.CrossRefGoogle Scholar
[17]Bingham, N. H. and Ostaszewski, A. J. Homotopy and the Kestelman–Borwein–Ditor Theorem. Canadian Bull. Math., to appear.Google Scholar
[18]Bingham, N. H. and Ostaszewski, A. J. Normed versus topological groups: dichotomy and duality. Dissertationes Math., to appear.Google Scholar
[19]Bingham, N. H. and Ostaszewski, A. J. Beyond Lebesgue and Baire II: Bitopology and measure-category duality. Colloq. Math., to appear.Google Scholar
[20]Bingham, N. H. and Ostaszewski, A. J. Topological regular variation. Cambridge Tracts in Math. (Cambridge University Press), in preparation.Google Scholar
[21]Borwein, D. and Ditor, S. Z.Translates of sequences in sets of positive measure. Canad. Math. Bull. 21 (1978), 497–498.Google Scholar
[22]Bhaskara Rao, K. P. S. and Reid, J. D.Abelian groups that are unions of proper subgroups. Bull. Austral. Math. Soc. 45 (1992), no. 1, 1–7.Google Scholar
[23]Bradford, J. C. and Goffman, C.Metric spaces in which Blumberg's theorem holds. Proc. Amer. Math. Soc. 11 (1960), 667–670.Google Scholar
[24]Christensen, J. P. R.On sets of Haar measure zero in abelian Polish groups. Israel J. Math. 13 (1972), 255–260 (1973).Google Scholar
[25]Ciesielski, K. and Pawlikowski, J.The covering property axiom, CPA. A combinatorial core of the iterated perfect set model, Cambridge Tracts in Math. 164 (Cambridge University Press, 2004).Google Scholar
[26]Comfort, W. W.Topological groups. Chapter 24, 1143–1263 in [52].CrossRefGoogle Scholar
[27]Csiszár, I. and Erdős, P.On the function g(t) = lim supx → ∞(f(x + t) − f(x)). Magyar Tud. Akad. Kut. Int. Közl. A 9 (1964), 603–606.Google Scholar
[28]Dales, H. G.Banach algebras and automatic continuity. London Math. Soc. Monog. 14 (Oxford University Press, 2000).Google Scholar
[29]Engelking, R.General Topology (Heldermann Verlag, 1989).Google Scholar
[30]Erdős, P.Set theoretic, measure theoretic, combinatorial, and number theoretic problems concerning point sets in Euclidean space. Real Anal. Exchange 4 (1978–79), 114–138.Google Scholar
[31]Fremlin, D. H.Measure-additive coverings and measurable selectors. Dissertationes Math. 260 (1987).Google Scholar
[32]Frolík, Z.Generalizations of the δ-property of complete metric spaces. Czechoslovak Math. J. 10 (85)(1960), 359–379.Google Scholar
[33]Fuchs, L.Infinite abelian groups, Vol. I. Pure and Applied Mathematics Vol. 36 (Academic Press, 1970).Google Scholar
[34]Ger, R. and Kuczma, M.On the boundedness and continuity of convex functions and additive functions. Aequationes Math. 4 (1970), 157–162.Google Scholar
[35]Graham, R. L., Rothschild, B. L. and Spencer, J. H.Ramsey Theory, 2nd ed. (Wiley, 1990) (1st ed. 1980).Google Scholar
[36]Halmos, P. R.Measure Theory (D. Van Nostrand, 1950) Graduate Texts in Math. 18 (Springer, 1979).CrossRefGoogle Scholar
[37]Hardy, G. H. and Wright, E. M.An introduction to the theory of numbers, Sixth edition. Revised by Heath-Brown, D. R. and Silverman, J. H. (Oxford University Press, 2008).Google Scholar
[38]Haworth, R. C. and McCoy, R. A.Baire spaces. Dissertationes Math. 141 (1977), 1–73.Google Scholar
[39]Heath, R. W. and Poerio, T. Topological groups and semi-groups on the reals with the density topology. Conference papers (Oxford, 2006) (Conference in honour of Peter Collins and Mike Reed).Google Scholar
[40]Hindman, N. and Strauss, D.Algebra in the Stone-Čech compactification. Theory and applications. de Gruyter Expositions in Mathematics, 27 (Walter de Gruyter & Co., 1998).CrossRefGoogle Scholar
[41]Hunt, B. R., Sauer, T., and Yorke, J. A.Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces. Bull. Amer. Math. Soc. 27·2 (1992), 217–238 (Addendum, Bull. Amer. Math. Soc. 28·2 (1993), 306–307).CrossRefGoogle Scholar
[42]Jayne, J. and Rogers, C. A.Analytic sets, part 1 of [85].Google Scholar
[43]Jensen, R. B.The fine structure of the constructible hierarchy. Ann. Math. Logic 4 (1972), 229–308.CrossRefGoogle Scholar
[44]Jen, R.The sumset phenomenon. Proc. Amer. Math. Soc. 130 (2002), no. 3, 855–861.Google Scholar
[45]Jones, F. B.Measure and other properties of a Hamel basis. Bull. Amer. Math. Soc. 48 (1942), 472–481.CrossRefGoogle Scholar
[46]Kechris, A. S.Classical descriptive set theory. Graduate Texts in Mathematics 156 (Springer, 1995).Google Scholar
[47]Kelly, J. C.Bitopological spaces. Proc. London. Math. Soc. 13 (1963), 71–89.Google Scholar
[48]Kestelman, H.The convergent sequences belonging to a set. J. London Math. Soc. 22 (1947), 130–136.Google Scholar
[49]Kestelman, H.On the functional equation f(x + y) = f(x) + f(y). Fund. Math. 34 (1947), 144–147.CrossRefGoogle Scholar
[50]Kominek, Z.On the sum and difference of two sets in topological vector spaces. Fund. Math. 71 (1971), no. 2, 165–169.CrossRefGoogle Scholar
[51]Kuczma, M.An introduction to the theory of functional equations and inequalities. Cauchy's functional equation and Jensen's inequality (PWN, Warsaw, 1985).Google Scholar
[52]Kunen, K. and Vaughan, J. E.Handbook of Set-Theoretic Topology (North-Holland Publishing Co., 1984).Google Scholar
[53]Kuratowski, K.Topology, Vol. I. (PWN, Warsaw 1966).Google Scholar
[54]Laczkovich, M.Analytic subgroups of the reals. Proc. Amer. Math. Soc. 126.6 (1998), 1783–1790.CrossRefGoogle Scholar
[55]Levi, S. On Baire cosmic spaces. General topology and its relations to modern analysis and algebra, V (Prague, 1981), 450–454, Sigma Ser. Pure Math. 3 (Heldermann, Berlin, 1983).Google Scholar
[56]Lukeš, J., Malý, J. and Zajiček, L.Fine topology methods in real analysis and potential theory. Lecture Notes in Mathematics, 1189 (Springer, 1986).CrossRefGoogle Scholar
[57]Mehdi, M. R.On convex functions. J. London Math. Soc. 39 (1964), 321–328.Google Scholar
[58]Michael, E.Almost complete spaces, hypercomplete spaces and related mapping theorems. Topology Appl. 41 (1991), no. 1–2, 113–130.Google Scholar
[59]van Mill, J.A note on the Effros Theorem. Amer. Math. Monthly 111.9 (2004), 801–806.CrossRefGoogle Scholar
[60]Miller, A. W.Infinite combinatorics and definability. Ann. of Pure and Appl. Math. Logic 41 (1989), 179–203 (see also updated web version at: http://www.math.wisc.edu/miller/res/).Google Scholar
[61]Miller, A. W.Descriptive Set Theory and Forcing (Springer, 1995).CrossRefGoogle Scholar
[62]Miller, A. W.Souslin's hypothesis and convergence in category. Proc. Amer. Math. Soc. 124 (1996), no. 5, 1529–1532.CrossRefGoogle Scholar
[63]Miller, A. W. and Muthuvel, K.Everywhere of second category sets. Real Anal. Exchange 24 (1998/99), no. 2, 607–614.Google Scholar
[64]Mostowski, A.Constructible sets with applications (PWN and North-Holland, 1969).Google Scholar
[65]Muthuvel, K.Application of translation of sets. Missouri J. Math. Sciences 3.3 (1991), 139–141.Google Scholar
[66]Muthuvel, K.Index of a subgroup of an abelian group. Missouri J. Math. Sciences, 13 (2001), no. 3, 187–194.CrossRefGoogle Scholar
[67]Muthuvel, K.Application of covering sets. Colloq. Math. 80 (1999), no. 1, 115–122.Google Scholar
[68]Namioka, I.Separate and joint continuity. Pacific J. Math. 51 (1974), 515–531.CrossRefGoogle Scholar
[69]Neumann, B.H.Groups covered by finitely many cosets. Publ. Math. Debrecen. 3 (1954/5), 227–242.Google Scholar
[70]Neumann, B. H.Groups coverd by permutable subsets. J. London Math. Soc. 29 (1954), 236–248.CrossRefGoogle Scholar
[71]Nowik, A. and Reardon, P.A dichotomy theorem for the Ellentuck topology. Real Anal. Exchange, 29 (2003/04), no. 2, 531–542.CrossRefGoogle Scholar
[72]Orlicz, W. and Ciesielski, Z.Some remarks on the convergence of functionals on bases. Studia Math. 16 (1958), 335–352.CrossRefGoogle Scholar
[73]Ostaszewski, A. J.On countably compact perfectly normal spaces. J. London Math. Soc. 14 (1976), 505–516.CrossRefGoogle Scholar
[74]Ostaszewski, A. J.Regular variation, topological dynamics, and the Uniform Boundedness Theorem. Topology Proc. 36 (2010), 305–336.Google Scholar
[75]Ostaszewski, A. J. Analytically heavy spaces: analytic Baire and analytic Cantor theorems, preprint.Google Scholar
[76]Ostaszewski, A. J. Beyond Lebesgue and Baire III: analyticity and shift-compactness, preprint.Google Scholar
[77]Ostaszewski, A. J. Continuity in groups: one-sided to joint, preprint.Google Scholar
[78]Ostaszewski, A. J. Analytic Baire spaces, preprint.Google Scholar
[79]Ostaszewski, A. J. On the Effros Open Mapping Principle, preprint.Google Scholar
[80]Ostrowski, A. Mathematische Miszellen XIV: Über die Funktionalgleichung der Exponentialfunktion und verwandte Funktionalgleichungen. Jahresb. Deutsch. Math. Ver. 38 (1929), 54–62, reprinted in Collected papers of Alexander Ostrowski, Vol. 4, 49–57 (Birkhäuser, Basel, 1984).Google Scholar
[81]Oxtoby, J. C.Measure and category, a survey of the analogies between topological and measure spaces, 2nd ed., Graduate Texts Math. 2 (Springer, 1980), (1st ed. 1971).Google Scholar
[82]Parthasarathy, K. R.Probability measures on metric spaces (AMS Chelsea Publishing RI, 2005), (reprint of the 1967 original).Google Scholar
[83]Pettis, B. J.Remarks on a theorem of E. J. McShane. Proc. Amer. Math. Soc. 2.1 (1951), 166–171.Google Scholar
[84]Piccard, S.Sur les ensembles de distances des ensembles de points d'un espace euclidien. Mém. Univ. Neuchâtel 13 (1939).Google Scholar
[85]Rogers, C. A., Jayne, J., Dellacherie, C., Topsøe, F., Hoffmann-Jørgensen, J., Martin, D. A., Kechris, A. S. and Stone, A. H.Analytic Sets (Academic Press, 1980).Google Scholar
[86]Rosendal, C. and Solecki, S.Automatic continuity of homomorphisms and fixed points on metric compacta. Israel J. Math. 162 (2007), 349–371.CrossRefGoogle Scholar
[87]Scheinberg, S.Topologies which generate a complete measure algebra. Adv. Math. 7 (1971), 231–239.CrossRefGoogle Scholar
[88]Shelah, S.Can you take Solovay's inaccessible away? Israel J. Math. 48 (1984), 1–47.CrossRefGoogle Scholar
[89]Solecki, S.Size of subsets of groups and Haar null sets. Geom. Funct. Anal. 15.1 (2005), 246–273.CrossRefGoogle Scholar
[90]Solecki, S. and Srivastava, S. M.Automatic continuity of group operations. Topology Appl. 77 (1997), 65–75.Google Scholar
[91]Solovay, R. M.A model of set theory in which every set of reals is Lebesgue measurable. Ann. of Math. 92 (1970), 1–56.CrossRefGoogle Scholar
[92]Steinhaus, H.Sur les distances des points de mesure positive. Fund. Math. 1 (1920), 93–104.Google Scholar
[93]Tao, T. and Vu, V. H.Additive Combinatorics. Cambridge studies in Advanced Mathematics 105 (2006).Google Scholar
[94]Topsøe, F. and Hoffmann-Jørgensen, J.Analytic spaces and their applications, Part 3 of [85].Google Scholar
[95]Trautner, R.A covering principle in real analysis. Quart. J. Math. Oxford (2), 38 (1987), 127–130.CrossRefGoogle Scholar
[96]Zapletal, J.Descriptive set theory and definable forcing. Mem. Amer. Math. Soc. 167 (2004), no. 793.Google Scholar