# Reference Request: Existence of Ordinal Rank Theory?

Notations: Recall that $\omega_1$ is the first uncountable ordinal. Let $X$ be a Polish space (completely metrizable and separable) and $F(X)$ be the collection of all real-valued functions on $X.$

A function $\rho:F(X)\to \omega_1$ is called an ordinal rank. In other words, ordinal rank assigns an ordinal to a function, typically measuring complexity of the function.

In $1990,$Kechris and Louveau conducted an investigation on three types of ordinal ranks acting on the class of Baire Class $1$ functions on a compact metric space (recall that $f$ is Baire Class $1$ if it is a pointwise limit of a sequence of continuous functions).

The three ranks are separation rank (introduced by Bourgain), oscillation rank and convergence rank (introduced by Zalcwasser, Gillespie and Hurwicz). They measure complexity of the respective definition of Baire Class $1$ functions (Baire Class $1$ functions have three equivalent definitions).

$26$ years later ($2016$), Elekes, Kiss and Vidnyánszky generalised from compact metric space to Polish space and the three ranks to Baire Class $\xi$ functions where $\xi$ is any countable ordinal (Recall that $f$ is Baire class $\xi$ if it is a pointwise limit of a sequence of Baire Class $\xi_n$ functions where $\xi_n<\xi$ for all natural number $n$). One of the generalisations is through topology refinement (For interested reader, please refer to section $5$ of their paper).

Question: Does there exist a literature on ordinal rank? More precisely, where can I find more information about ordinal rank?

• Kechris' 1995 book Classical Descriptive Set Theory has this in Chapter IV. You might also want to look over his Ph.D. students --- Ramsamujh's dissertation, in particular, studies certain ordinal rank functions on the set of continuous nowhere differentiable functions, and functions with everywhere divergent Fourier series (special case: those whose coefficients approach zero), and compact simply connected sets in the plane, among other things. – Dave L Renfro May 26 '18 at 15:52
• The link "Marton..." to the AMS paper goes through some library proxy. I suggest you link the abstract of the paper. By the way, those are the first names of the researchers. You would probably like to have their surnames there. – Pedro Sánchez Terraf May 27 '18 at 12:22
• @PedroSánchezTerraf Edited. Thanks for pointing out. – Idonknow May 27 '18 at 12:44
• @DaveLRenfro Unfortunately, my institution does not have access to Ramsamujh's dissertation. – Idonknow May 27 '18 at 13:08

This is a topic I briefly looked at in 1992-1993 (I was mostly interested in ranks for differentiable functions and for nowhere differentiable continuous functions), when I was working on my dissertation, and at that time I collected a few papers on the topic (and some more throughout the 1990s). However, even though I have a “designated notebook binder” for papers on this topic (which I stopped adding papers to around 1999), I've never posted a list of such papers, or even made such a list for my own personal use, so I suppose this is something I can do today since today is a holiday for me.

Below is a chronological list of items from my $\leq$ 1990s collection of stuff. The dates at the end of each entry are submission dates or book preface dates, which I included to make it easier for me to chronologically order the items. I had planned to include all relevant items after the 1990s, but I found way too much by googling, so I’ll leave the more recent work for you to find. There does not seem to be any survey/expository paper on ordinal ranks in general (however, I’ve included 3 items from after the 1990s that seem useful in this regard), so this is something you might want to consider writing if you wind up getting very engrossed in this topic.

Regarding searching online, besides googling (variations of) the authors’ names, the following phrases, when googled, will lead you to more than I’m willing to look at. For some of these phrases, such as “convergence rank”, you will need to include “Kechris” as an additional search term.

Lusin-Sierpinski index [or Luzin-Sierpinski index], Piatetski-Shapiro rank, Zalcwasser rank, Szlenk index, Denjoy rank, Kechris-Woodin rank, oscillation rank, convergence rank, differentiability rank. Also, try googling “coanalytic rank” and (as a separate search) “co-analytic rank”.

[1] Miklós Ajtai and Alexander Sotirios Kechris, The set of continuous functions with everywhere convergent Fourier series, Transactions of the American Mathematical Society 302 #1 (July 1987), 207-221. [11 June 1985]

[2] Alexander Sotirios Kechris and William Hugh Woodin, Ranks of differentiable functions, Mathematika 33 #2 (December 1986), 252-278. [1 July 1985]

[3] Taje Indralall Ramsamujh, Some Topics in Descriptive Set Theory and Analysis, Ph.D. Dissertation (under Alexander Sotirios Kechris), California Institute of Technology, 1986, vi + 147 pages. abstract [5 May 1986]

[4] Taje Indralall Ramsamujh, A comparison of the Jordan and Dini tests, Real Analysis Exchange 12 #2 (1986-1987), 510-515. [correction in RAE 14 #1 (1988-1989), pp. 251–252] [11 November 1986]

[5] Alexander Sotirios Kechris and Alain Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness, London Mathematical Society Lecture Notes Series #128, Cambridge University Press, 1987, viii + 367 pages. [June 1987]

[6] Alexander Sotirios Kechris and Russell David Lyons, Ordinal rankings on measures annihilating thin sets, Transactions of the American Mathematical Society 310 #2 (December 1988), 747-758. [10 September 1987]

[7] Taje Indralall Ramsamujh, The complexity of nowhere differentiable continuous functions, Canadian Journal of Mathematics 41 #1 (1989), 83-105. [22 October 1987]

[8] Alexander Sotirios Kechris and Alain Louveau, A classification of Baire class $1$ functions, Transactions of the American Mathematical Society 318 #1 (March 1990), 209-236. [15 December 1987]

[9] Alexander Sotirios Kechris, Alain Louveau, and Valérie Tardivel [Tardivel-Nachef], The class of synthesizable pseudomeasures, Illinois Journal of Mathematics 35 #1 (Spring 1991), 107-146. [22 November 1988]

[10] Taje Indralall Ramsamujh, Three ordinal ranks for the set of differentiable functions, Journal of Mathematical Analysis and Applications 158 #2 (1 July 1991), 539-555. [25 August 1989]

[11] Alexander Sotirios Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics #156, Springer-Verlag, 1995, xviii + 402 pages. corrections and updated [September 1994]

An introduction to ordinal ranks is given in Chapter V.1 (pp. 140-150), and several ordinal ranks are defined and used in the remainder of the book.

[12] Haseo Ki, The Kechris-Woodin rank is finer than the Zalcwasser rank, Transactions of the American Mathematical Society 347 #11 (November 1995), 4471-4484. [14 September 1994]

Ordinal ranks are discussed in Sections 34-36 (pp. 267-312).

[13] Haseo Ki, Topics in Descriptive Set Theory Related to Number Theory and Analysis, Ph.D. Dissertation (under Alexander Sotirios Kechris), California Institute of Technology, 1995, v + 72 pages. abstract [15 March 1995]

[14] Haseo Ki, On the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank, Transactions of the American Mathematical Society 349 #7 (July 1997), 2845-2870. [13 April 1995]

[15] Taje Indralall Ramsamujh, Simply connected compact subsets of the plane, Journal of Mathematical Analysis and Applications 237 #1 (1 September 1999), 240-252. [19 August 1998]

[16] Spiros A. Argyros, Gilles Godefroy, and Haskell Paul Rosenthal, Descriptive set theory and Banach spaces, pp. 1007-1069 in Johnson/Lindenstrauss (editors), Handbook of the Geometry of Banach Spaces, Volume 2, Elsevier Science B.V., 2003, xii + 1007-1866 pages.

[17] Yiannis Nicholas Moschovakis, Descriptive Set Theory, 2nd edition, Mathematical Surveys and Monographs #155, American Mathematical Society, 2009, xiv + 502 pages.

See Section 2D (pp. 62-64).

[18] Zoltán Vidnyánszky, Descriptive Set Theoretical Methods and Their Applications, Ph.D. Dissertation (under Márton Elekes), Eötvös Loránd University, 2014, 112 + 2 pages.