ROBERT F. TICHY: 50 YEARS —THE UNREASONABLE EFFECTIVENESS
OF A NUMBER THEORISTHANSJÖRG ALBRECHER, MICHAEL DRMOTA, MARTIN GOLDSTERN,PETER J. GRABNER, AND REINHARD WINKLER

Figure 1. Climbing in the Gesäuse.
The present volume of UDT is devoted to Robert F. Tichy on the occasion of
his 50th birthday. We cordially congratulate him on this occasion and wish him
the best for the future.
In this short note we collect highlights of his scientific work. Together with morethan 70 coauthors he has written over 200 papers so far, with topics that range fromnumber theory to applications in actuarial mathematics and also mathematicalchemistry.
But, of course, the scientific work is only one facet of Robert Tichy’s personality. It is remarkable how many students graduated under his supervision since1984. At least 25 PhD students are known to us, including the five authors of the†The title alludes to E. Wigner’s 1960 article “The unreasonable effectiveness of mathematics
in the natural sciences”, and to S. A. Burr’s 1992 book “The unreasonable effectiveness of number
theory”. Both touch aspects of mathematics and number theory which are in Robert Tichy’s
spirit.
1

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H. ALBRECHER, M. DRMOTA, M. GOLDSTERN, P. GRABNER, AND R. WINKLER

present article. The majority of his former students are still active researchers inmathematics, 9 with habilitation, three of them in the position of a full professor.
Such quantities, though very impressive, do not explain the individuality of an
academic personality. We remember the time when we were students and Robert
Tichy was a very young docent of less than 30 years in Vienna and then, with 33,
professor in Graz. He was one of those academic teachers who regularly offeredspecial courses on many diverse and challenging mathematical topics. They were
popular not only because of the interesting content but also because of Robert
Tichy’s qualities as an excellent and inspiring lecturer with a very characteristic
and appealing sense of humour. Some of his courses were closely connected to his
own very diverse mathematical research. So he attracted a considerable part of theinterested students of our generation to scientific activity in mathematics.
As Tichy’s students (and coauthors) we are always impressed by his sense forgood problems and his intuition how to tackle them. None of the problems heposed were unsolvable. But for each of them one had to study and further develop
advanced methods from analysis and other parts of mathematics, thus becoming
familiar with new and substantial parts of mathematics. At the same time one had
the chance to develop one’s own mathematical personality by emphasising favourite
aspects of a given topic. In face of his extraordinary drive and energy Robert Tichy
responds to the growth of young mathematical personalities sensitively and in avery inspiring way. He knows the strengths and weaknesses of each of his students.
Thus, whenever he suggested a research topic to one of us, we still could be sure thatthe topic would fit perfectly to this person’s individual mathematical taste and, duealso to Robert Tichy’s clear-sighted mathematical judgement and taste for fruitfulproblems, that the topic would open up new perspectives. Particularly motivating
for his students is his astonishing ability to combine high exigence with humanunderstanding – aspects of personality which often seem contradictory. Anotherpleasant result of this individual mentoring is the amicable atmosphere among his
former students, where individuality counts more than competition.
Anybody who knows Robert Tichy as a researcher and as a teacher will find it
even more unbelievable that he finds the time for his extremely effective activitiesin science administration: head of department, faculty dean, president of the Austrian Mathematical Society and representative of pure mathematics in the Austrian
Science Foundation FWF. Even more impressing: Whenever he meets one of us hetakes the time for a personal chat, maybe about his favourite leisure occupation:
mountaineering. It is a matter of interpretation whether this affection for nature,physical activity and overcoming difficulties is a compensation for his mathematicalpassion or part of it.
Definitely part of his mathematical passion are uniform distribution, digital expansions, Diophantine equations and actuarial mathematics. We add some remarks
on Robert Tichy’s work in these areas.
1. Uniform Distribution Modulo 1
Robert Tichy’s first major scientific interest was the theory of uniformly distributed sequences. After his PhD thesis “Gleichverteilung von Mehrfachfolgen undKetten” (University of Vienna, 1979) under the supervision of Edmund Hlawka andmotivated by the study of the monographs by Kuipers–Niederreiter and Hlawka hesubstantially contributed to this theory and several of his very first PhD students(Gerhard Turnwald, Michael Drmota, Martin Goldstern, Reinhard Winkler, Peter

ROBERT F. TICHY: 50 YEARS

3

Grabner) wrote their theses in this area. It is impossible to summarise all his contributions in uniform distribution theory. We thus concentrate only on one aspect,
namely on metric discrepancy theory. He also worked, for example, on special sequences, on generalised notions of discrepancy, on uniformly distributed functions
and on discrete versions of the theory. Further, together with Michael Drmota healso wrote a Lecture Notes volume “Sequences, Discrepancies and Applications” [9]that covers the 20 years development of the theory after Kuipers’ and Niederreiter’smonograph from 1974. It has become a standard reference in this field.
It is well known that the theory of uniformly distributed sequences is closely
related to that of random numbers. In volume 2 of the monograph series “The artof computer programming” D. Knuth had posed the problem whether there exists
a real number x > 1 such that the sequence xan is u.d. mod 1 for all algorithmicallycomputable sequences an of different natural numbers. This problem was solved byNiederreiter and Tichy [23] (and then substantially generalised by Tichy [32]) by
proving that almost all x > 1 have this property. In [32] Tichy also shows that the1
discrepancy DN of this sequence is bounded above by C · N − 2 +η (for arbitrarily
small η > 0). Later Losert, Nowak and Tichy [22] considered powers of matrices andproved corresponding¡ ¢nresults. One should also mention that it is still an unsolved
problem whether 32 or en are u.d. mod 1.
The situation is a little bit different if one considers sequences of the form αx n .
For example, if x = q ≥ 2 is an integer then the sequence αq n is u.d. mod 1 if and√only if α is normal with respect to base q. Although it is unknown whether 2,
π or e are normal it is relatively easy to provide explicit normal numbers; e.g. theChampernowne number α = 0.123456789101112131415 . . . is normal in base 10.
More generally sequences of the kind αan have been considered. For example,
H. Weyl proved in his fundamental paper from 1916 that for any increasing sequence an of integers it follows that αan is u.d. mod 1 for almost all real α. Later
J.W.S. Cassels and P. Erdős-J. F. Koksma proved independently that the discrep5
1ancy can be bounded by C · N − 2 (log N ) 2 +η . Another highlight was W. Philipp’s
1977 result on lacunary sequences, that is, an+1 ≥ (1 + ρ)an with ρ > 0. He showedthat for almost all α(1.1)

1
N DN (αan )≤ lim sup √≤ C.
4N log log NN →∞Recently, Berkes, Philipp and Tichy [7] provided a very strong generalisation of(1.1) for sequences an that are only weakly increasing in the sense thatak+k1−η ≥ k ak

(k ≥ k0 (η))

for some η > 0. Interestingly, the authors also have to assume two Diophantine
conditions on an . First, the number of solutions (h, n) with 1 ≤ h ≤ R of theequation han = b shall not exceed C Rγ for some γ < 21 and all natural numbers b.
Second, the number of solutions (n1 , n2 , n3 , n4 ) with 1 ≤ ni ≤ N of the equation
h1 an1 + h2 an2 + h3 an3 + h4 an4 = 0 shall not exceed C0 N 1+β for some β < 12 and for
all fixed integers hi with 0 < |hi | ≤ N 3 provided that there are no proper subsums
that vanish. The proof of this result uses sophisticated methods from probability
theory such as martingale embeddings of empirical processes. Furthermore it is aninteresting number-theoretic question to show that the Diophantine conditions are
satisfied for special sequences. For example, one has to use recent versions of thesubspace theorem of J.-H. Evertse, H. P. Schlickewei and W. Schmidt to verify that4

H. ALBRECHER, M. DRMOTA, M. GOLDSTERN, P. GRABNER, AND R. WINKLER

the so-called Hardy-Littlewood-Pólya sequence an , that is, the sequence q1e1 · · · qrer
arranged in increasing order where qj are coprime integers, meets these properties.
2. Digital Expansions and Number Theory
In the late 1980s Robert Tichy became interested in number theoretical problems
related to digital expansions. This led to a series of contributions to various aspectsof this subject and to three PhD-theses on related topics.
It was observed by a French school of mathematicians around J. Coquet, H. Delange, H. Faure, P. Liardet, M. Mendès-France, G. Rauzy, G. Rhin, and manyothers that digital representations of the positive integers and additive functions
given by these representations can be used to construct uniformly distributed sequences. Ergodic theory gives a different point of view on such constructions, which
serve as interesting and diverse examples for dynamical systems.
Motivated by the study of uniformly distributed sequences his first contribution
was a detailed study of the discrepancy of the sequence (αsq (n))n∈N for irrational αin [21], where sq (n) denotes the q-adic sum-of-digits function. Best possible upperand lower bounds for the discrepancy as well as the uniform discrepancy of suchsequences depending on Diophantine approximation properties of α were also given.
Digital functions such as the binary sum-of-digits are not only of number-theoretical interest, but also occur naturally in the context of average case analysis ofalgorithms. In [10] the classical Mellin-Perron formula was used to derive H. Delange’s summation formula for the sum-of-digits function and other digital functions
in a unified way. Publication in a Computer Science journal made this a standard
reference for such type of results and techniques.
Digital representations with respect to a given increasing sequence of integers
G = (Gn )n∈N0 can be used to construct “adic” compactifications KG of the positive
integers. The addition-by-one map τ acts naturally on KG . In [15], Grabner,
Liardet and Tichy studied the dynamical system (KG , τ ) from a topological andmeasure theoretic point of view, and in particular investigated the continuity of τand measure theoretic isomorphisms to group rotations. Continuing earlier work of
J. C. Alexander and D. B. Zagier on the entropyof Bernoulli convolutions related√
1+ 5
to representations of real numbers to base 2 , in [14] a description of this entropy
in terms of combinatorics on words could be given. In a series of papers T. Kamae
developed several techniques for proving spectral disjointness of dynamical systems
related to the sum-of-digits functions sp (n) and sq (n) with respect to coprime bases
p and q. In [16] finite automata and M. Queffélec’s point of view based on Šreı̆dercharacters are used to derive spectral disjointness for skew-products given by moregeneral additive functions with respect to multiplicatively independent bases.
The study of the arithmetic structure of sets defined by congruence relations
on the sum-of-digits function or more general additive digital functions has been
initiated by A. O. Gelfond and continued by É. Fouvry, C. Mauduit, J. Rivat, and
A. Sárközy. The paper [28] contributes to these investigations by proving an ErdősKac type theorem for the distribution of the arithmetic function ω(n), where n is
restricted by congruence relations for several additive digital functions with respect
to pairwise coprime bases. A very involved combination of the Hardy-Littlewood
circle method with sieve techniques was used in [29] to count the number of solutionsof the equation
N = xk1 + · · · + xks

ROBERT F. TICHY: 50 YEARS

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under the restriction that sq (xi ) ≡ h (mod m) (for i = 1, . . . , s), i.e. Waring’s
problem with digital restrictions.
3. Diophantine Equations
Many Diophantine approximation problems are closely related to distributional
problems of special sequences. For example, the order of the discrepancy of the
Weyl sequence αn depends on the continued fraction expansion of α that encodes
the approximation behaviour of α by rationals. It is a long standing problem to finda corresponding property for the Kronecker sequence (α1 , . . . , αk )n, for example, interms of Diophantine approximation conditions for αj . It was therefore a natural
step for Tichy to work on Diophantine problems and — as it turned out — on
Diophantine equations, too.
He worked, for example, on parametrised families of Thue equations [18, 20,19], where one can use Alan Baker’s theory of linear forms in the logarithms ofalgebraic numbers. For several examples, Tichy and his coauthors could determine
all solutions. It is a highly non-trivial problem to provide a complete solution of a
Diophantine equation in 2 variables and 1 parameter; usually two variables cause
enough troubles. Even if one knows that there are only finitely many solutions of
the equation F (x, y) = 0 it is difficult (and mostly impossible) to determine all ofthem.
It is therefore a remarkable result that Bilu and Tichy [8] gave a complete anddefinite answer to the equation f (x) = g(y), where f and g are rational polynomials.
Before partial results have been obtained by M. D. Fried and A. Schinzel. Theirmain result says that the equation f (x) = g(y) has infinitely many rational solutions(with a bounded denominator) if and only if there are linear polynomials λ(x),µ(x) ∈ Q[x], a polynomial ϕ(x) ∈ Q[x] and a pair (f1 , g1 ) in an explicit set of five
families of pairs of polynomials (including, for example, Dickson polynomials) with
f = ϕ ◦ f1 ◦ λ,g = ϕ ◦ g1 ◦ µ

and the equation f1 (x) = g1 (y) has infinitely many rational solutions with abounded denominator. The proof rests on arguments of Fried and Schinzel but
includes several new ideas. One main point is to determine when the polynomial
f (x) − g(y) has an exceptional factor, that is, when there exists an irreducible polynomial F (x, y) ∈ Q[x, y] which divides f (x) − g(y) and defines a plane curve of
genus zero and with at most two points at infinity. This result can be applied, forexample, to Diophantine equations of the shape f (x) = g(y), where f and g depend
on unknown parameters, that is, we have again more than two variables.
Another series of Tichy’s papers is related to Diophantine problems of linear
recurrence sequences. In [13] he provided together with C. Fuchs (under suitable
and natural conditions) an explicit upper bound for the number of solutions (x, n)of the equation Gn = xq , where Gn is a linear recurrence sequence and q > 1 a given
integer. Two further papers with Fuchs and Pethő [11, 12] deal with recurrences
Gn+d (x) = Ad−1 (x)Gn+d−1 (x) + · · · + A0 (x)Gn (x),for n ≥ 0,of polynomials Gn (x), Aj (x) ∈ K[x], where K is a field of characteristic 0. They
consider the Diophantine equation
Gn (x) = Gm (P (x))
and obtain an upper bound for the number of solutions (m, n) that depends only
on d.

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H. ALBRECHER, M. DRMOTA, M. GOLDSTERN, P. GRABNER, AND R. WINKLER

4. Actuarial Mathematics
In the early 80’s, Robert Tichy got into contact with some problems from actuarial risk theory that deal with the stochastic modelling of the free reserve in a portfolio of insurance contracts. He quickly realized that rigorous analytical techniques
could be employed to develop new solution procedures for ruin-related quantities
and, in turn, these practical problems can also trigger theoretical research questions. Many equations in this field are of renewal type and Robert Tichy not only
contributed criteria for existence and uniqueness of corresponding solutions, butalso developed a rather versatile numerical solution method based on Quasi-Monte
Carlo techniques (cf. [30]), in that way linking risk theory with uniform distribution
theory. Later on, this method was extended to more general risk models in [4].
Exact solutions of partial integro-differential equations occurring in risk theory
are rather scarce, mostly due to rather unpleasant boundary conditions. However,such exact expressions are extremely helpful since they allow to tune model parameters towards a given target for ruin-related quantities, such as ensuring a certain
degree of solvency. In this connection, Robert Tichy introduced Laplace transform
methods to the field of linear dividend barrier problems [31] and together withSiegl in [24] considerably enlarged the available set of exact solutions, developinga recursive solution algorithm (for which a rigorous convergence proof could thenbe given in Albrecher and Tichy [6]; further extensions of that method to arbitrary
moments of expected dividend payments and discounted penalty functions were recently derived in [3]). Further objects of study in this context range from intricate
recurrences (for the exact solution of finite-time ruin probabilities under interest
force [5]) to certain types of differential equations with delayed arguments, that
occur when implementing securitization strategies in the risk model [26, 27].
Among his most frequently cited papers in this field is [25], where additional
stochastic factors in the dividend risk model were introduced and where it was
shown that the toolkit of analytical solution procedures carries over in a transparent
way, an approach that was subsequently taken up by several other research groups
for further investigation.
Robert Tichy also contributed to the development of efficient Quasi-Monte Carlo
methods for pricing products that bridge insurance activities with the financial
market, such as catastrophe bonds [1, 2]. Such pricing problems sometimes leadto integrals with singular integrands, and the suitability of Quasi-Monte Carlotechniques in these situations was then investigated in [17].
Being himself a fully qualified actuary, Robert Tichy has a lot of insight in practical issues of the insurance industry, which is an ideal complement to his profoundknowledge of mathematics. In this respect, he recently also played an instrumental role in installing a Master program in Financial and Actuarial Mathematics inGraz, which will offer future students an entry to this fascinating field of appliedmathematics.
Gaudeat igitur, iuvenis dum est!

ROBERT F. TICHY: 50 YEARS

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References[1] Hansjörg Albrecher, Jürgen Hartinger, and Robert F. Tichy, Multivariate approximation
methods for the pricing of catastrophe-linked bonds, Internat. Ser. Numer. Math. 145 (2003),
21–39.
, Quasi-Monte Carlo techniques for CAT bond pricing, Monte Carlo Methods & Appl.
[2](2004), no. 10, 197–212.
[3] Hansjörg Albrecher, Jürgen Hartinger, and Robert F. Tichy, On the distribution of dividend
payments and the discounted penalty function in a risk model with linear dividend barrier,
Scand. Actuar. J. (2005), no. 2, 103–126.
[4] Hansjörg Albrecher, Reinhold Kainhofer, and Robert F. Tichy, Simulation methods in ruinmodels with non-linear dividend barriers, Math. Comput. Simulation 62 (2003), 277–287.
[5] Hansjörg Albrecher, Jozef Teugels, and Robert F. Tichy, On a gamma series expansion for
the time-dependent probability of collective ruin, Insurance: Mathematics and Economics 29(2001), no. 3, 345–355.
[6] Hansjörg Albrecher and Robert F. Tichy, On the convergence of a solution procedure for arisk model with gamma-distributed claims, Schweiz. Aktuarver. Mitt. 2 (2000), 115–127.
[7] István Berkes, Walter Philipp, and Robert F. Tichy, Empirical processes in probabilistic
number theory: the LIL for the discrepancy of (nk ω) mod 1, Illinois J. Math. 50 (2006),
no. 1-4, 107–145 (electronic).
[8] Yuri F. Bilu and Robert F. Tichy, The Diophantine equation f (x) = g(y), Acta Arith. 95(2000), no. 3, 261–288.
[9] Michael Drmota and Robert F. Tichy, Sequences, discrepancies and applications, Lecture
Notes in Mathematics, vol. 1651, Springer-Verlag, Berlin, 1997.
[10] Philippe Flajolet, Peter Grabner, Peter Kirschenhofer, Helmut Prodinger, and Robert F.
Tichy, Mellin transforms and asymptotics: digital sums, Theoret. Comput. Sci. 123 (1994),
no. 2, 291–314.
[11] Clemens Fuchs, Attila Pethő, and Robert F. Tichy, On the Diophantine equation G n (x) =Gm (P (x)), Monatsh. Math. 137 (2002), no. 3, 173–196.
[12], On the Diophantine equation Gn (x) = Gm (P (x)): higher-order recurrences, Trans.
Amer. Math. Soc. 355 (2003), no. 11, 4657–4681 (electronic).
[13] Clemens Fuchs and Robert F. Tichy, Perfect powers in linear recurring sequences, Acta Arith.
107 (2003), no. 1, 9–25.
[14] Peter J. Grabner, Peter Kirschenhofer, and Robert F. Tichy, Combinatorial and arithmeticalproperties of linear numeration systems, Combinatorica 22 (2002), no. 2, 245–267, Special
issue: Paul Erdős and his mathematics.
[15] Peter J. Grabner, Pierre Liardet, and Robert F. Tichy, Odometers and systems of numeration,
Acta Arith. 70 (1995), no. 2, 103–123.
, Spectral disjointness of dynamical systems related to some arithmetic functions,[16]
Publ. Math. Debrecen 66 (2005), no. 1-2, 213–243.
[17] Jürgen Hartinger, Reinhold Kainhofer, and Robert F. Tichy, Quasi-Monte Carlo algorithms
for unbounded, weighted integration problems, J. Complexity 20 (2004), no. 5, 654–668.
[18] Clemens Heuberger, Attila Pethő, and Robert F. Tichy, Complete solution of parametrized
Thue equations, Proceedings of the 13th Czech and Slovak International Conference on Number Theory (Ostravice, 1997), vol. 6, 1998, pp. 93–114.
, Thomas’ family of Thue equations over imaginary quadratic fields, J. Symbolic[19]
Comput. 34 (2002), no. 5, 437–449.
[20] Clemens Heuberger and Robert F. Tichy, Effective solution of families of Thue equations
containing several parameters, Acta Arith. 91 (1999), no. 2, 147–163.
[21] Gerhard Larcher and Robert F. Tichy, Some number-theoretical properties of generalized
sum-of-digit functions, Acta Arith. 52 (1989), no. 2, 183–196.
[22] Viktor Losert, Werner-Georg Nowak, and Robert F. Tichy, On the asymptotic distributionof the powers of s × s-matrices, Compositio Math. 45 (1982), no. 2, 273–291.
[23] Harald Niederreiter and Robert F. Tichy, Solution of a problem of Knuth on complete uniform
distribution of sequences, Mathematika 32 (1985), no. 1, 26–32.
[24] Thomas Siegl and Robert F. Tichy, Lösungsmethoden eines Risikomodells bei exponentiell
fallender Schadensverteilung, Schweiz. Aktuarver.Mitt. (1996), no. 1, 85–118.
[25], A process with stochastic claim frequency and a linear dividend barrier, Insurance
Math. Econom. 24 (1999), no. 1-2, 51–65, 1st IME Conference (Amsterdam, 1997).

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H. ALBRECHER, M. DRMOTA, M. GOLDSTERN, P. GRABNER, AND R. WINKLER[26], Ruin theory with risk proportional to the free reserve and securitization, Insurance
Math. Econom. 26 (2000), no. 1, 59–73.
[27], A model in ruin theory using derivative securities, Mitt. Schweiz. Aktuarvereinigung(2002), no. 1, 13–30.
[28] Jörg M. Thuswaldner and Robert F. Tichy, An Erdős-Kac theorem for systems of q-additive
functions, Indag. Math. (N.S.) 11 (2000), no. 2, 283–291.
, Waring’s problem with digital restrictions, Israel J. Math. 149 (2005), 317–344,[29]
Probability in mathematics.
[30] Robert F. Tichy, Über eine zahlentheoretische Methode zur numerischen Integration und zur
Behandlung von Integralgleichungen, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber.
II 193 (1984), no. 4-7, 329–358.
[31], Bemerkung zu einem versicherungsmathematischen Modell, Mitt. Verein. Schweiz.
Versicherungsmath. (1987), no. 2, 237–241.
, Ein metrischer Satz über vollständig gleichverteilte Folgen, Acta Arith. 48 (1987),[32]
no. 2, 197–207.
(H. Albrecher) Radon Institute, Austrian Academy of Sciences, Altenbergerstrasse
69, 4040 Linz, Austria
E-mail address: hansjoerg.albrecher@oeaw.ac.at(M. Drmota, M. Goldstern, R. Winkler) Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Vienna,
AustriaE-mail address: michael.drmota@tuwien.ac.at
Martin.Goldstern@tuwien.ac.at
reinhard.winkler@tuwien.ac.at(P. Grabner) Institut für Analysis und Computational Number Theory, Technische
Universität Graz, Steyrergasse 30, 8010 Graz, Austria
E-mail address: peter.grabner@tugraz.at