Welcome
At the Vieux Port in Marseilles, France, Sept. 13, 2013
I am Senior Scientist at the University of Leoben (Austria). My research is centered around, but not restricted to, the base-q representation of an integer. The three main manifestations of this topic in my research concern (0) linear subsequences of substitutive sequences, for example, of the Thue–Morse sequence; (1) understanding addition in terms of digital expansions; (2) divisibility of combinatorially interesting numbers like binomial coefficients. I like other beautiful things too, such as finding mathematics in nature (see below), or piano music.
Research
Preprints
| 2. | Lukas Spiegelhofer | Thue–Morse along the sequence of cubes | [arXiv] |
| 1. | Bartosz Sobolewski and Lukas Spiegelhofer | Block occurrences in the binary expansion | [arXiv] |
Peer reviewed papers
| 28. | Jean-Marc Deshouillers, Michael Drmota, Clemens Müllner, Andrei Shubin, and Lukas Spiegelhofer | Synchronizing automatic sequences along Piatetski-Shapiro sequences Israel J. Math., to appear | [arXiv] |
27. | Michael Drmota, Clemens Müllner, and Lukas Spiegelhofer | Primes as sums of Fibonacci numbers (135 pages) Accepted for publication in Mem. Amer. Math. Soc. | [ arXiv | web ] |
| 26. | Lukas Spiegelhofer | Collisions of digit sums in bases 2 and 3 Israel J. Math. 258 (2023), no. 1, 475–502 | [ arXiv | web ] |
| 25. | Lukas Spiegelhofer | Gaps in the Thue–Morse word J. Aust. Math. Soc. 114 (2023), no. 1, 110–144 | [ arXiv | web ] |
24. | Myriam Amri, Lukas Spiegelhofer, and Jörg Thuswaldner | Répartition jointe dans les classes de résidus de la somme des chiffres pour deux représentations d'Ostrowski Int. J. Number Theory 18 (2022), no. 5, 955–976 | [ arXiv | web ] |
| 23. | Lukas Spiegelhofer and Michael Wallner | The binary digits of n+t Ann. Sc. Norm. Super. Pisa Cl. Sci. 24 (2023), no. 1, 1–31 | [ arXiv | web ] |
| 22. | Michael Drmota, Christian Mauduit, Joël Rivat, and Lukas Spiegelhofer | Möbius orthogonality for sequences with maximal entropy J. Anal. Math. 146 (2022), no. 2, 531–548. | [ arXiv | web ] |
| 21. | Lukas Spiegelhofer | A lower bound for Cusick's conjecture on the digits of n+t Math. Proc. Cambridge Philos. Soc 172 (2022), no. 1, 139–161 | [ arXiv | pdf ] |
| 20. | Lukas Spiegelhofer | The level of distribution of the Thue–Morse sequence Compos. Math. 156 (2020), no. 12, 2560-2587 | [ arXiv | web ] |
| 19. | Lukas Spiegelhofer and Thomas Stoll | The sum-of-digits function on arithmetic progressions Mosc. J. Comb. Number Theory 9 (2020), no. 1, 43-49 | [ arXiv | Sage ] |
| 18. | Lukas Spiegelhofer | Approaching Cusick's conjecture on the sum-of-digits function Integers 19 (2019), Paper No. A59 | [ arXiv ] |
| 17. | Jean-Marc Deshouillers, Michael Drmota, Clemens Müllner, and Lukas Spiegelhofer | Randomness and non-randomness properties of Piatetski-Shapiro sequences modulo m Mathematika 65 (2019), no. 4, 1051–1073 | [ arXiv ] |
| 16. | Lukas Spiegelhofer and Jeffrey Shallit | Continuants, run lengths, and Barry's modified Pascal triangle Electron. J. Combin. 26 (2019), no. 1, Paper 1.31 | [ arXiv ] |
| 15. | Lukas Spiegelhofer and Michael Wallner | The Tu–Deng conjecture holds almost surely Electron. J. Combin. 26 (2019), no. 1, article P1.28 | [ arXiv ] |
| 14. | Sandro Bettin, Sary Drappeau, and Lukas Spiegelhofer | Statistical distribution of the Stern sequence Comment. Math. Helv. 94 (2019), no. 2, 241–271 | [ arXiv ] |
| 13. | Lukas Spiegelhofer and Michael Wallner | Divisibility of binomial coefficients by powers of two J. Number Theory 192 (2018), 221–239. | [ arXiv ] |
| 12. | Lukas Spiegelhofer | Discrepancy results for the Van der Corput sequence Unif. Distrib. Theory 13 (2018), no. 2, 57-69 | [ arXiv ] |
| 11. | Michael Drmota, Clemens Müllner, and Lukas Spiegelhofer | Möbius orthogonality for the Zeckendorf sum-of-digits function Proc. Amer. Math. Soc. 146 (2018), no. 9, 3679–3691. | [ arXiv ] |
| 10. | Lukas Spiegelhofer | Pseudorandomness of the Ostrowski sum-of-digits function J. Théor. Nombres Bordeaux 30 no. 2 (2018), 637-649 | [ arXiv ] |
| 9. | Lukas Spiegelhofer | A digit reversal property for an analogue of Stern's sequence J. Integer Seq. 20 (2017) | [ arXiv ] |
| 8. | Lukas Spiegelhofer | A digit reversal property for Stern polynomials INTEGERS 17 (2017), Paper No. A53, 7 pp. | [ arXiv ] |
| 7. | Lukas Spiegelhofer and Michael Wallner | An explicit generating function arising in counting binomial coefficients divisible by powers of primes Acta Arith. 181 (2017), no. 1, 27-55 | [ arXiv | Abstract | web ] |
| 6. | Clemens Müllner and Lukas Spiegelhofer | Normality of the Thue–Morse sequence along Piatetski-Shapiro sequences, II Israel J. Math. 220 (2017), no. 2, 691–738 | [ arXiv | Abstract | BibTeX | web ] |
| 5. | Michael Coons and Lukas Spiegelhofer | The maximal order of hyper-(b-ary) expansions Electron. J. Combin. 24 (2017), no. 1, Paper 1.15 | [ arXiv | Abstract | BibTeX | web ] |
| 4. | Michael Drmota, Manuel Kauers, and Lukas Spiegelhofer | On a conjecture of Cusick concerning the sum of digits of n and n+t SIAM J. Discrete Math. 30 (2016), no. 2, 621–649. | [ arXiv | Abstract | BibTeX | web ] |
| 3. | Lukas Spiegelhofer | Normality of the Thue–Morse sequence along Piatetski-Shapiro sequences Q. J. Math. 66 (2015), no. 4, 1127–1138. | [ arXiv | Abstract | BibTeX | web ] |
| 2. | Lukas Spiegelhofer | Piatetski-Shapiro sequences via Beatty sequences Acta Arithmetica 166 (2014), no. 3, 201–229. | [ arXiv | Abstract | BibTeX | web ] |
| 1. | Johannes Morgenbesser and Lukas Spiegelhofer | A reverse order property of correlation measures of the sum-of-digits function INTEGERS 12 (2012), Paper No. A47, 5 pp. | [ PDF | Abstract | BibTeX ] |
Other publications
| 1. | Michael Coons and Lukas Spiegelhofer | Number theoretic aspects of regular sequences. Sequences, groups, and number theory, 37–87, Trends Math., Birkhäuser/Springer, Cham, 2018. |
Theses
| 3. | Lukas Spiegelhofer | Subsequences of digitally defined functions Habilitationsschrift, TU Wien, 2023 | [PDF] |
| 2. | Lukas Spiegelhofer | Correlations for numeration systems PhD thesis written under the joint supervision of Michael Drmota and Joël Rivat, TU Wien and Aix-Marseille Université, 2014. | [PDF] |
| 1. | Lukas Spiegelhofer | Universal properties and categories of modules Diploma thesis written under the supervision of Johannes Schoißengeier, Universität Wien, 2011. | [PDF] |
Talks
Thue–Morse along the sequence of cubes Invited, March 2024 | Slides (PDF) |
Thue–Morse along the sequence of cubes Combinatorics on words, CIRM, Marseille, France; Invited, February 2024 | Slides (PDF) |
The simplest 2-regular sequences JKU Linz, December 2023 | Slides (PDF) Zoom into the minimum of a fractal function encoding the number of odd binomial coefficients |
Subsequences of digitally defined functions IMath Webinar of the University of the Philippines Diliman, Invited, November 2023. | Slides (PDF) |
Subsequences of digitally defined functions Habilitationskolloquium, TU Wien, October 2023 | Slides (PDF) |
Thue–Morse along the sequence of cubes September 2023. | Slides (PDF) |
Primes as sums of Fibonacci numbers Teletalk at One World Numeration Seminar, October 2022 | |
Primes as sums of Fibonacci numbers II ArithRand 2022, TU Graz, Austria, July 2022 | Slides (PDF) |
Subsequences of digitally defined functions Talk given in the course of obtaining the postdoctoral teaching qualification, TU Wien, Austria, March 2022 | Slides (PDF) |
Collisions of digit sums in different bases Rencontres de théorie analytique et élémentaire des nombres, IHP Paris, Invited, March 2022 | Slides (PDF) |
The level of distribution of the Thue–Morse sequence, Rencontres de théorie analytique et élémentaire des nombres, IHP Paris, Invited, March 2022 | Slides (PDF) |
The base-2 expansion along arithmetic progressions Teletalk at the Departmentseminar, Montanuniversität Leoben, Austria, March 2021. | Slides (PDF) |
The digits of n+t Teletalk at One World Numeration Seminar December 2020. | Slides (PDF) |
The digits of n+t Teletalk at Séminaire Ernest Marseille, France, October 2020. | Slides (PDF) |
Digital expansions along arithmetic progressions, Montanuniversität Leoben, Austria, March 2020. | Slides (PDF) |
Möbius orthogonality and the sum-of-digits function Numeration and Substitution, Vienna, Austria, July 2019. | Slides (PDF) |
The level of distribution of the Zeckendorf sum of digits Topology of planar and higher dimensional self-replicating tiles, Salzburg, Austria, February 2019.
| Slides (PDF) |
The level of distribution of the Thue–Morse sequence Numeration 2018, Paris, France, May 2018. | |
The level of distribution of the Thue–Morse sequence Kooperationsworkshop SFB55, Linz Austria, April 2018 | Slides (PDF) |
Divisibility of binomial coefficients by powers of primes ELAZ 2016, Strobl, Austria, September 2016. | |
Divisibility of binomial coefficients by powers of primes Numeration 2016, Prague, Czech Republic, May 2016. | |
Divisibility of binomial coefficients by powers of two Séminaire de Théorie des Nombres de Nancy–Metz, Nancy, France, March 2016. | |
Divisibility of binomial coefficients by powers of two Arbeitsgemeinschaft Diskrete Mathematik, TU Wien, Austria, January 2016. | |
Divisibility of binomial coefficients by powers of two Zahlentheoretisches Kolloquium, TU Graz, Austria, January 2016. | |
On a problem by Cusick concerning the sum of digits of n and n+t | |
Normality of the Thue–Morse sequence along Piatetski-Shapiro sequences II Numeration 2015, Nancy, France, May 2015. | |
Piatetski-Shapiro sequences via Beatty sequences Pseudorandomness in Number Theory, CIRM (Marseille, France), July 2014. | |
| The sum of digits of n and n+t, Arbeitsgemeinschaft Diskrete Mathematik, TU Wien, Austria, June 2014. | |
Approximating Piatetski-Shapiro sequences by Beatty sequences Séminaire de Théorie des Nombres de Nancy–Metz, Nancy, France, Invited, February 2014. |
FWF Project
In June 2022 my four-year FWF project "Correlations of k-regular and morphic sequences" was approved.
Currently, Bartosz Sobolewski is employed as a postdoc in this project, and Pascal Jelinek as a predoc.
Honors and awards
- Edmund und Rosa Hlawka-Preis für Mathematik" 2022 by the Austrian Academy of Sciences.
- Joint paper "On a conjecture of Cusick concerning the sum of digits of n and n+t", with Drmota and Kauers, published in SIAM J. Discrete Math., won the best paper award 2017 of the Faculty of Mathematics and Geoinformation, TU Vienna
Press coverage:
Editorial duties
I am an editor for Math. Pannon.
Short CV
- 12/2023 Postdoctoral teaching qualification (Habilitation) at the TU Wien
- 9/2022–present Senior Scientist at Montanuniversität Leoben
- 10/2020–8/2022 University assistant at Montanuniversität Leoben
- 02/2017–9/2020 Postdoc at the Vienna University of Technology
- 09/2016–01/2017 Postdoc at the JKU Linz, Austria
- 02/2016–08/2016 Postdoc at the Université de Lorraine, France
- 01/2015–01/2016 Postdoc at the Vienna University of Technology
- 2011–2014 PhD studies of mathematics at the Vienna University of Technology and the Aix–Marseille Université under the supervision of Michael Drmota and Joël Rivat.
- 2005–2011 Studies of mathematics at the University of Vienna
- 2004–2005 Studies of physics at the Vienna University of Technology (TU Wien)
Trivia
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