Preprints and Papers by Christian Elsholtz

This page contains a list of papers that appeared, are submitted or are still unfinished. My Habilitationsschrift (Post Doc Thesis) combines methods from number theory and combinatorics. Amongst others it presents a new version of the large sieve method which has applications to gaps of primes. (See for example the paper No. 4 The inverse Goldbach problem below)
Back to my Homepage.

  Qualification Theses
The Erdos-Straus conjecture Diplomarbeit (Master Thesis equivalent)
Sums of k unit fractions Ph.D. Thesis (parts appeared in Transactions of the AMS)
Combinatorial Prime Number Theory Habilitationsschrift (Postdoc Thesis). (External referees: Brüdern (Stuttgart), Heath-Brown (Oxford), A. Sárközy (Budapest). Internal referees: Lucht, Hilgert).

  Title Abstract Author(s) Status Download
1 Primzahlen der Form $p=4k+1$ sind Summe von zwei Quadratzahlen   Christian Elsholtz Mathematik Lehren. (Februar 1994, Heft 62, 58-61) pdf
2 A remark on Hofmann and Wolke's additive decompositions of the set of primes.   Christian Elsholtz Arch. Math. 76, No.1, 30-33 (2001). Link to publisher Springer Link pdf
3 Sums of $k$ unit fractions   Christian Elsholtz Trans. Amer. Math. Soc. 353 (2001), no. 8, 3209-3227. Transactions of the AMS dvi ps pdf
4 The inverse Goldbach problem   Christian Elsholtz Mathematika 48 (2001), 151-158 dvi ps pdf scanned pdf
5 The distribution of sequences in residue classes   Christian Elsholtz Proc. Amer. Math. Soc. 130 (2002), no.8, 2247-2250 Proceedings of the AMS dvi ps pdf
6 Some remarks on the additive structure of the set of primes   Christian Elsholtz Number theory for the millennium, I (Urbana, IL, 2000), 419--427, (Proceedings of the Millennial Conference on Number Theory (Bennett et.al.), AK Peters, 2002). ps pdf
7 Upper bounds for prime $k$-tuples of size $\log N$ and oscillations   Christian Elsholtz Arch. Math. 82 (2004), 33-39. Link to publisher ps pdf
8 Lower bounds for multidimensional zero sums   Christian Elsholtz Combinatorica 24, (2004) no. 3, 351 - 358. Link to publisher dvi ps pdf
9 Triples of primes in arithmetic progressions   Christian Elsholtz Quart. J. Math. 53 (2002), 393-395. Link to publisher pdf ps
10 The Liouville---Heath-Brown---Zagier proof of the two squares theorem


Kombinatorische Beweise des Zweiquadratesatzes und Verallgemeinerungen
  Christian Elsholtz The english version contains all details of the proofs. The German version has a more general introduction and is directed towards a general audience and contains another short proof. The German version is published in Mathematische Semesterberichte 50, Heft 1, 77-93, 2003. English preprint. ps

German version pdf
Link to publisher
11 Zeroes of $L$-functions attached to cuspidal representation of congruence subgroups of $GL_n (\mathbb{Z})$   Christian Elsholtz, Jacek Pomykala, Warsaw submitted, Preprint  
12 On primes of the form $x^2+y^4$   Christian Elsholtz submitted, Preprint  
13 On cluster primes   Christian Elsholtz Acta Arithmetica, 110 (2003), 281-284. Link to publisher dvi ps pdf
14 Von Lücken zwischen Primzahlen zur Optimierung von Gitterpunkten. Survey for a general audience (in German) Christian Elsholtz invited paper by TU Contact, magazine of the TU Clausthal, issue 12, 2003, pages 61-64 1) zum Heft
2) pdf
3) internet links that are mentioned in the paper
15 A short survey on upper and lower bounds for multidimensional zero sums This survey appeared with years of delay. For an update see "Zero-sum problems in finite abelian groups and affine caps" below. Christian Elsholtz Springer Lecture Notes in Computer Science, LNCS 4123, pages 1079-1080, 2006.   pdf
Link to the publisher
16 On variants of the larger sieve   Ernie Croot, Georgia Tech., Christian Elsholtz Acta Mathematica Hungarica 103 (3) (2004) 243-254 Link to publisher ps pdf
17 On Thin Sets of Primes Expressible as Sumsets   Ernie Croot, Georgia Tech., Christian Elsholtz Acta Mathematica Hungarica, 106 (3), 2005, 197 - 226 Link to publisher ps pdf
18 A simple proof of Sen's possibility theorem on majority decisions   Christian Elsholtz, C. List, LSE, London Elemente der Mathematik 60 (2005), 45-56. Link to publisher pdf
19 Shifted products that are coprime pure powers   Rainer Dietmann, Stuttgart; Christian Elsholtz, Katalin Gyarmati and Miki Simonovits, Budapest Journal of combinatorial Theory, Series A, 111 (1), 2005, 24-36. Link to publisher ps pdf
20 Maximal dimension of unit simplices   Christian Elsholtz, Walter Klotz Discrete Comput. Geom. 34 (2005), no. 1, 167--177. Link to publisher dvi ps pdf
21 Zero-sum problems in finite abelian groups and affine caps   Yves Edel, Christian Elsholtz, Alfred Geroldinger, Silke Kubertin, Laurence Rackham Quarterly Journal of Mathematics 58 (2007), 159-186 Link to publisher pdf
22 Additive decomposability of multiplicatively defined sets   Christian Elsholtz This paper was invited for an issue at the occasion of the 75th birthday of Prof. Wirsing. Functiones et Approximatio 35 (2006), 61-77. Link to publisher
pdf
23 Sudoku im Mathematikunterricht   Christian Elsholtz, A. Mütze Mathematische Semesterberichte 54 (1), 2007, 69-93 Springerlink:
pdf

An extended version is posted
here. Also the internetlinks can be reached from here.
24 The number Gamma(k) in Waring's problem   Christian Elsholtz Acta Arith. 131 (2008), 43-49 pdf Acta Arithmetica
25 Multiplicative decomposability of shifted sets   Christian Elsholtz Bulletin of the London Mathematical Society 2008, 40: 97-107 pdf publisher's site
26 Sums of two squares and one biquadrate   Rainer Dietmann, Christian Elsholtz Funct. Approx. Comment. Math. Volume 38, Number 2 (2008), 233-234. pdf publisher's site
27 Non-simple abelian varieties in a family: geometric and analytic approaches   J.Ellenberg (Madison, WI), Christian Elsholtz, C.Hall (U o Michigan, Ann Arbor), E.Kowalski (ETH Zürich) Journal of the LMS 2009 80(1), 135-154 Arxiv, Link to Publisher
28 Independence and interdependence in collective decision making: an agent-based model of nest-site choice by honeybee swarms   Christian Elsholtz, C.List (LSE), T.D.Seeley (Cornell) Phil. Trans. R. Soc. B Volume 364, Number 1518 / March 27, 2009 Publisher, pdf
Press coverage:
Daily Telegraph,
Economist, Economist
BBC Radio Wales, Friday 13th February 2009, 4.56pm (Good Evening Wales)
Recommended by F1000 biology
The paper is top ten in the most downloaded of the journal in 2009.
I am mentioned (as a mathematician, albeit not by name) in the cover story of Science News: Swarm Savvy How bees, ants and other animals avoid dumb collective decisions By Susan Milius May 9th, 2009; Vol.175 #10 (p. 16)
Science News Cover
29 A survey on additive and multiplicative decompositions of sumsets and of shifted sets   Christian Elsholtz Chapter in "Combinatorial Number Theory and Additive Group Theory" Series: Advanced Courses in Mathematics - CRM Barcelona, Geroldinger, Alfred, Ruzsa, Imre Z. 2009, Birkhäuser, (based on the DocCourse on Additive Combinatorics, Barcelona 2008). Pages, 213-231. Publisher pdf,
Publisher
30 A combinatorial approach to sums of two squares and related problems   Christian Elsholtz Invited paper, appeared in: Additive Number Theory: Festschrift in Honor of the sixtieth birthday of Melvyn B. Nathanson, Springer Verlag, 2010, pages 115-140. pdf,
link to publisher
31 Hilbert cubes in progression-free sets and in the set of squares   Rainer Dietmann, Christian Elsholtz Israel Journal of Mathematics, November 2012, Volume 192, Issue 1, pp 59-66 link to publisher
32 Prime divisors of thin sequences   Christian Elsholtz The American Mathematical Monthly, Vol. 119, No. 4 (April 2012), pp. 331-333 Link to publisher (JSTOR)
A major part of this paper was kindly included in "Proofs from the BOOK", by M. Aigner and G. Ziegler, (in the first chapter: Six proofs of the infinity of primes, with an appendix on infinitely many more proofs) Link to publisher (Springer), needs subscription.
33 The number of representations of rationals as a sum of unit fractions Printing took a while (printed in 2013!), but IJM may be one of the few journals, which allows the author to use the final(!) version for private/homepages/print servers. Tim Browning, Christian Elsholtz Illinois J of Mathematics 55 (2), 2011, 685-696. pdf
34 Monochromatic solutions of linear equations.   Christian Elsholtz, Frank Weber submitted  
35 Patterns and complexity of multiplicative functions   Yvonne Buttkewitz, Christian Elsholtz Journal of the London Mathematical Society 84 (3) 2011, 578-594. Publisher
36 A problem of Ramanujan, Erdos and Katai on the iterated divisor function   Yvonne Buttkewitz, Christian Elsholtz, Kevin Ford, Jan-Christoph Schlage-Puchta IMRN 2012, issue 17, 4051-4061. arxiv,
free final version, with permission of OUP
37 Egyptian fractions with restrictions   Yong-Gao Chen, Christian Elsholtz, and Li-Li Jiang (Nanjing, China) Acta Arith. 154 (2012), 109-123 arxiv, publisher,
38 The central limit theorem for subsequences in probabilistic number theory   Christoph Aistleitner, Christian Elsholtz Canad. J. Math. 64(2012), 1201-1221 publisher,
39 Counting the number of solutions to the Erdos-Straus equation on unit  fractions   Christian Elsholtz, Terence Tao J. Aust. Math. Soc. 94 (2013), 50–105 arxiv Link to publisher
40 The number of Huffman codes, compact trees, and sums of unit fractions   Christian Elsholtz, Clemens Heuberger, Helmut Prodinger IEEE Transactions of Information Theory 59 (2) 2013, 1065-1075. arxiv IEEE (final version free)
41 On Gaps Between Primitive Roots in the Hamming Metric   Rainer Dietmann, Christian Elsholtz, Igor Shparlinski Quarterly J. Math. 64 (no 4), 2013, 1043-1055 arxiv,
publisher's link publisher's link 2
42 An Alternative Proof on Four-Dimensional Zero-Sums   Christian Elsholtz Papers in Number Theory, RMS-Lecture Notes Series No. 23, 2016, pp. 29–36. pdf
43 On Romanov's constant   Christian Elsholtz, JC Schlage-Puchta Mathematische Zeitschrift publisher ,
44 Sumsets being squares   Andrej Dujella, Christian Elsholtz Acta Math. Hungar. 141 (2013), no. 4, 353–357. publisher , final publisher pdf version, with kind permission of the publisher
45 On Gaps Between Quadratic Non-residues in the Euclidean and Hamming Metrics   Rainer Dietmann, Christian Elsholtz, Igor Shparlinski Indag. Math. (N.S.) 24 (2013), no. 4, 930–938. publisher
46 On Diophantine quintuples and D(-1)-quadruples   Christian Elsholtz, Alan Filipin, Yasutsugu Fujita Monatsh. Math. 175 (2014), 227-239. publisher pdf
47 Congruence properties of multiplicative functions on sumsets and monochromatic solutions of linear equations   Christian Elsholtz, David S. Gunderson Funct. Approx. Comment. Math.Volume 52, Number 2 (2015), 263-281. publisher ,
48 Almost all primes have a multiple of small Hamming weight   Christian Elsholtz Bull. Australian Math. Soc. 94 (2) 2016, 224-235. publisher, arxiv
49 Additive decompositions of sets with restricted prime factors   Christian Elsholtz, Adam J. Harper Trans. Amer. Math. Soc. 367 (2015) 7403-7427. pdf arxiv
50 Hilbert cubes in arithmetic sets   Rainer Dietmann, Christian Elsholtz Revista Matemática Iberoamericana Revista Matemática Iberoamericana 31 (4), 2015, 1477–1498. arxiv publisher
51 On conjectures of T. Ordowski and Z.W. Sun concerning primes and quadratic forms   Christian Elsholtz, Glyn Harman In: Springer (eds. C. Pomerance and M. Rassias): Analytic Number Theory (In Honor of Helmut Maier’s 60th Birthday), (2015), 65-81 Springer link pdf
52 Sums of two squares and a power   Rainer Dietmann, Christian Elsholtz in book: From Arithmetic to Zeta-Functions - Number Theory in Memory of Wolfgang Schwarz, Editors: Sander, Jürgen, Steuding, Jörn, Steuding, Rasa, Springer, 2016, pages 103-108. Springer link, arxiv
53 Prescribing the binary digits of squarefree numbers and quadratic residues   Rainer Dietmann, Christian Elsholtz, Igor Shparlinski To appear in Transactions of the AMS arxiv publisher
54 The curvature of sequences moderately dense in the primes   Jörg Brüdern, Christian Elsholtz in book: Number Theory – Diophantine Problems, Uniform Distribution and Applications, Festschrift in Honour of Robert F. Tichy’s 60th Birthday, C. Elsholtz, P. Grabner (editors), Springer 2017. pages 193-210. arxiv Springer
55 Golomb's conjecture on prime gaps   Christian Elsholtz The American Mathematical Monthly April 2017, 365-368 arxiv jstor
56 Title (Schur)   Christian Elsholtz, Laurence Rackham submitted  
57 On Erdős and Sárközy's sequences with Property P   Christian Elsholtz, Stefan Planitzer Monatshefte für Mathematik 182, Issue 3, 2017, 565–575 Springer link (free download)
58 Egyptian fractions with odd denominators   Christian Elsholtz Q. J. Math. 67, No. 3, 425-430 (2016). arxiv Link to publisher
59 Maximal sumfree sets of integer lattice grids   Christian Elsholtz, Laurence Rackham J London Mathematical Society 95(2), 2017, 353–372. Link to publisher
60 On the regularity of primes in arithmetic progressions   Christian Elsholtz, Niclas Technau, Robert Tichy Int. J. Number Theory 13 (2017), no. 5, 1349–1361. arxiv Link to publisher
61 Romanov   Christian Elsholtz, Florian Luca, Stefan Planitzer submitted
62 Number Theory – Diophantine Problems, Uniform Distribution and Applications Festschrift in Honour of Robert F. Tichy’s 60th Birthday, Presenting 22 research and survey papers written by leading experts in their respective fields, it focuses on areas around R.F. Tichy's interests, including Diophantine problems, asymptotic counting, uniform distribution and discrepancy of sequences (in theory and application), dynamical systems, prime numbers, and actuarial mathematics. (Editors:) Christian Elsholtz, Peter Grabner Number Theory – Diophantine Problems, Uniform Distribution and Applications Festschrift in Honour of Robert F. Tichy’s 60th Birthday, Editors: Christian Elsholtz, Peter Grabner, 2017, Springer. Springer
63 Euler   Rainer Dietmann, Christian Elsholtz submitted
64 The maximal order of iterated multiplicative functions   Christian Elsholtz, Marc Technau, Niclas Technau submitted

Papers on prime numbers: 1,2,4,6,7,9,10,12,13,14,17,22,25,29,30,32,39 42,43,48,51,54,55,60,61,63,64
Papers on sumsets: 2,4,6,8,12,15,17,21,22,24,26,29,31,42,43,44,47,48,49,50,51,52,53,57,59,61
Papers using sieve methods: 2,3,4,5,6,7,13,16,17,22,25,27,29,49,50,54,61
Papers on arithmetic functions: 35, 36, 39,47,57,64
Papers on zero sums: 8,15,21,42,48
Papers on diophantine equations, Waring's problem: 3,24,26,33, 34, 37, 39,40, 46,52,58,63
Papers on quadratic nonresidues etc: 31,41,45,53
Papers in combinatorics: 8,9,14,15,18,19,20,21,31,34,35,47,50,56,59
Papers in algebra: 20,21,27
Papers in probabilistic number theory: 38
Papers on mathematical modelling: 28
Papers for general audience (e.g. teachers): 1,10,14,18,23,28,30,32,55

Papers in 2001: 2,3,4
Papers in 2002: 5,6,9
Papers in 2003: 10,13,14
Papers in 2004: 7,8,16
Papers in 2005: 17,18,19,20
Papers in 2006: 15,22
Papers in 2007: 21,23
Papers in 2008: 24,25,26
Papers in 2009: 27,28,29
Papers in 2010: 30
Papers in 2011: 33,35
Papers in 2012: 31,32,36,37,38
Papers in 2013: 39,40,41,44,45
Papers in 2014: 46
Papers in 2015: 47,49,50,51
Papers in 2016: 42, 48, 52, 58
Papers in 2017+: 43, 53,54,55,57,59,60,61,63,64
Key publications include:
3. Sums of $k$ unit fractions. Trans. Amer. Math. Soc. 353 (2001), no. 8, 3209--3227.2)
4. The inverse Goldbach problem. Mathematika 48 (2001), 151-158.
8. Lower bounds for multidimensional zero sums. Combinatorica 24 (2004), no. 3, 351 -358.
21. Zero-sum problems in finite abelian groups and affine caps, (with Y. Edel (Heidelberg), A. Geroldinger (Graz), S. Kubertin, L. Rackham), Quarterly J. Math. (Oxford) 58 (2007), 159-186.
25. The multiplicative decomposability of shifted sets, Bulletin of the London Mathematical Society 40 (2008), 97-107.
27. Non-simple abelian varieties in a family: geometric and analytic approaches, with J. Ellenberg (Wisconsin, Madison), C. Hall (U of Michigan), E. Kowalski (ETH Zurich). J. London Math. Soc. (2) 80 (2009), 135-154.
28. Independence and interdependence in collective decision-making: an agent-based model of nest-site choice by honey bee swarms, with C. List (LSE), T.D. Seeley (Cornell), Phil. Trans. R. Soc. B Volume 364, Number 1518 / March 2009.
31. Hilbert cubes in progresson-free sets and in the set of squares, with Rainer Dietmann, Israel Journal of Mathematics 192 (2012), 192 (1), pp 59-66
35. Patterns and complexity of multiplicative functions, with Yvonne Buttkewitz, Journal of the London Mathematical Society 84 (3) 2011, 578-594.
36. A problem of Ramanujan, Erdos and Katai on the iterated divisor function, with Yvonne Buttkewitz, Kevin Ford, Jan-Christoph Schlage-Puchta, IMRN 2012, issue 17, 4051-4061.
39. Counting the number of solutions to the Erdos-Straus equation on unit fractions, with Terence Tao
40. The number of Huffman codes, compact trees, and sums of unit fractions, with Clemens Heuberger, Helmut Prodinger
41. On Gaps Between Primitive Roots in the Hamming Metric, with Rainer Dietmann, Igor Shparlinski, Quarterly J. Math.
43. On Romanov's constant, with JC Schlage-Puchta.
49. Additive decompositions of sets with restricted prime factors, with Adam Harper.
50. Hilbert cubes in arithmetic sets, with Rainer Dietmann
53. Prescribing the binary digits of squarefree numbers and quadratic residues, with Rainer Dietmann, Igor Shparlinski
54. The curvature of sequences moderately dense in the primes, with J Brudern
59. Maximal sumfree sets of integer lattice grids