| 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) | ||
| 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 |
23 | Sudoku im Mathematikunterricht | Christian Elsholtz, A. Mütze | Mathematische Semesterberichte 54 (1), 2007, 69-93 |
Springerlink: 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
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. That chapter "Six proofs ..." is also included in "Pi und Co.: Kaleidoskop der Mathematik" (editors: Ehrhard Behrends, Peter Gritzmann, Günter M. Ziegler.) |
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. | 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, |
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. | 43 | On Romanov's constant | Christian Elsholtz, JC Schlage-Puchta | Mathematische Zeitschrift (2018) 288:713-724 | publisher , 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. | arxiv publisher, | 49 | Additive decompositions of sets with restricted prime factors | Christian Elsholtz, Adam J. Harper | Trans. Amer. Math. Soc. 367 (2015) 7403-7427. | arxiv publisher | 50 | Hilbert cubes in arithmetic sets | Rainer Dietmann, Christian Elsholtz | 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. | arxiv publisher, | 53 | Prescribing the binary digits of squarefree numbers and quadratic residues | Rainer Dietmann, Christian Elsholtz, Igor Shparlinski | Trans. Amer. Math. Soc. 369 (2017), 8369-8388 | 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 124. no 4, 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 type problems | Christian Elsholtz, Florian Luca, Stefan Planitzer | The Ramanujan journal 47 (2018), Issue 2, 267-289. | Link to publisher (open access) Link to publisher | 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 | Sums of two numbers having only prime factors congruent to one modulo four | This asymptotically solves an old problem of Euler. | Rainer Dietmann, Christian Elsholtz |
Banach Center Publications 118 (2019), 177-179 Number theory week 2017, Banach centre publications, vol 118, Institute of mathematics, Polish academy of Sciences, 2019, editors: Łukasz Pankowski and Maciej Radziejewski | pdf publisher | 64 | The maximal order of iterated multiplicative functions | Christian Elsholtz, Marc Technau, Niclas Technau | Mathematika, 65 (4) 2019, 990-1009 | arxiv , publisher | 65 | The number of solutions of the Erdős-Straus Equation and sums of k unit fractions | Christian Elsholtz, Stefan Planitzer | Proceedings of the Royal Society of Edinburgh Section A, 150(3) (2020), 1401--1427. | arxiv publisher | 66 | Unconditional Prime-representing Functions, Following Mills | Christian Elsholtz | American Mathematical Monthly 127(7) (2020), 639-642. | arxiv, publisher | 67 | The density of integers representable as the sum of four prime cubes | Christian Elsholtz, Jan-Christoph Schlage-Puchta | Acta Arithmetica 192 (2020), 363-369 | arxiv publisher | 68 | Longer gaps between values of binary quadratic forms | Rainer Dietmann, Christian Elsholtz | (not published, extended version see paper 77) | arxiv | 69 | Arithmetic progressions in binary quadratic forms and norm forms | Christian Elsholtz, Christopher Frei | Bulletin London Math. Soc. 51(4), 2019, 595-602. | arxiv, publisher (open access) | 70 | Algorithmic counting of nonequivalent compact Huffman codes | Christian Elsholtz, Clemens Heuberger, Daniel Krenn | Applicable Algebra in Engineering, Communication and Computing (2023) | arxiv doi (open access) | 71 | Caps and progression-free sets in Z_m^n | Christian Elsholtz, Peter Pal Pach | Designs, Codes and Cryptopgraphy 88 (2020), 2133-2170. | journal (html, open access), journal (pdf, open access) arxiv | 72 | Fermat's Last Theorem implies Euclid's infinity of the primes | This paper contains (inter alia): FLT and Schur's theorem implies Euclid's theorem.
In other words: the existence of infinitely many primes is a *necessary*
condition for Fermat's Last Theorem to hold. A number of other examples are given,
including a quite simple graph theoretic lemma. This paper also contains
several new proofs of Euclid's theorem: 1) Fermat, with fixed exponent 3 or 4, can (as far as I can see) be proven without Euclid's theorem, so this gives a new proof. 2) Lemma 3 and section 4 (with finitely many primes the set of k-th powers would have positive density, a contradiction) 3) Lemma 6 and Theorem 5 (every finitely coloured complete graph contains a certain complete monochromatic bipartite subgraph, and some simple number theory). |
Christian Elsholtz | American Mathematical Monthly 128:3, 250-257, 2021. Many thanks to Timothy Gowers for advertising this paper, part 1 part 2 The paper was also highlighted by the Mathematical Association of America (MAA), by the Global Head of Portfolio at the publisher Taylor and Francis. Many thanks also to Richard Lipton and Kenneth Regan for reviewing the paper on their blog Gödel's Lost letter. Bill Gasarch wrote a similar paper, including actually the same proof (FLT and Schur implies Euclid), with more discussions from a computer science/complexity theory background, and applying it also to rings. He also discusses the paper on his blog. Quanta magazine covered the new proof. (A correction to the Quanta article: it suggests that Gasarch's proof and my proof are distinct, while they are the same.) |
publisher, open access, | 73 | Sums of four and more unit fractions and approximate parametrizations | Christian Elsholtz, Stefan Planitzer | Bulletin London Math. Soc. 53 (3), 2021, 695-709. | publisher (open access), | 74 | Exponentially Larger Affine and Projective Caps | Christian Elsholtz, Gabriel Lipnik | Mathematika Volume 69 (2023), Issue 1 p. 232-249 | publisher arxiv, publisher (open access). | 75 | Large Subsets of Z_m^n without Arithmetic Progressions | Christian Elsholtz, Benjamin Klahn, Gabriel Lipnik | Designs, codes and cryptography Volume 91, pages 1443–1452, (2023) | arxiv. doi (open access). | 76 | On polynomials with roots modulo almost all primes | Christian Elsholtz, Benjamin Klahn, Marc Technau | Acta Arithmetica 205 (2022), 251-263 | arxiv. publisher. | 77 | Longer gaps between values of binary quadratic forms (replaces and extends 68) | Rainer Dietmann, Christian Elsholtz, Alexander Kalmynin, Sergei Konyagin, James Maynard | International Mathematics Research Notices (IMRN), Volume 2023, Issue 12, June, Pages 10313–10349. | IMRN (doi). pdf of accepted version. | 78 | PS primes | Christian Elsholtz, Jan-Christoph Schlage-Puchta | submitted | 79 | Some results of James Maynard | invited survey for the IMN (2022) | Christian Elsholtz, Robert Tichy | IMN | 80 | Egyptian fractions | invited survey, for a wide audience, but with some number theoretic background | Thomas Bloom, Christian Elsholtz | Nieuw Archief voor Wiskunde, 23(4), December 2022, 237-245. | arxiv, journal | 81 | Maximal line-free sets in F_p^n | Christian Elsholtz, Jakob Führer, Erik Füredi, Benedek Kovács, Péter Pál Pach, Dániel Simon, Nóra Velich | Periodica Math Hungarica (open access) | 82 | Sumsets in the set of squares | Christian Elsholtz, Lena Wurzinger | Quarterly Journal of Mathematics (open access) | (83) | (New lower bounds for three-term progression free sets in F_n^p) | Christian Elsholtz, Laura Proske, Lisa Sauermann | arxiv, (This manuscript was merged with the paper below and will not be published in this form.) | 84 | Improving Behrend's construction: Sets without arithmetic progressions in integers and over finite fields | Christian Elsholtz, Zach Hunter, Laura Proske, Lisa Sauermann | arxiv, |
Papers on prime numbers: 1,2,4,6,7,9,10,12,13,14,17,22,25,29,30,32,39 42,43,48,51,54,55,60,61,63,64,65,66, 67, 68,72,78
Papers on sumsets: 2,4,6,8,12,15,17,21,22,24,26,29,31,42,43,44,47,48,49,50,51,52,53,57,59,61,71,84
Papers using sieve methods: 2,3,4,5,6,7,13,16,17,22,25,27,29,49,50,54,61,84
Papers on arithmetic functions: 35, 36, 39,47,57,63,64,65,66,70,73,77
Papers on zero sums: 8,15,21,42,48,71
Papers on multidimensional problems: 8,15,20, 21,42,48,71,74,75,81,83,84
Papers on arithmetic progressions (in the sense of additive combinatorics):
9,31,34,35,(37),50,53,54,69,71,72,74,75,81, 83,84
Papers on diophantine equations, including Waring's problem:
3,24,26,33, 34, 37, 39,40, 46,52,58,63,65,70,72,80
Papers on sums of unit fractions (Egyptian fractions): 3, 33,37,39,40,58,65,70,73,80
Papers on Waring type problems: 24,26,48, 52,57,63, 67,77
Papers on quadratic nonresidues etc: 31,41,45,48,53,84
Papers in combinatorics: 8,9,14,15,18,19,20,21,31,34,35,47,50,56,59,70,71,72,74,75, 83,84
Papers in algebra: 20,21,27,69, 76,77
Papers in probabilistic number theory: 38,77
Papers on mathematical modelling: 28
Papers for general audience (e.g. teachers): 1,10,14,18,23,28,30,32,55,66,72,79, 80
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: 53,54,55,57,59,60, Book: 62
Papers in 2018: 43,61
Papers in 2019: 63,64,69
Papers in 2020: 65,66,67,71
Papers in 2021: 72,73
Papers in 2022: 76, 79, 80
Papers in 2023: 70, 74,75,77
Papers in 2024+: 78, 81,82,84
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, with L. Rackham
65. The number of solutions of the Erdős-Straus Equation and sums of k unit fractions, with S. Planitzer
71. Caps and progression-free sets in Z_m^n, with P.P. Pach
74. with G. Lipnik
77. with Dietmann, Kalmynin, Konyagin, Maynard
84. Improving Behrend's construction: Sets without arithmetic progressions
in integers and over finite fields (Elsholtz, Hunter, Proske, Sauermann)
A couple of external Theses study some of the work above in detail.
Here are some:
Master's Thesis at Zagreb University (2002) On sums of two squares
Link
Master's Thesis at Simon Fraser University (2016)
Agent-based modelling of nest site selection by bees and ants
Link
