Steven J. Miller                                                      
Associate Professor of Mathematics, Williams College               
202 Bronfman Science Center, 413-597-3293
(Steven.J.Miller AT williams.edu)
 
[Click here for my schedule]    [Click here for general advice]

curriculum vita (click here for annotated bibliography)
research statement       and     teaching statement
Takeaways (all classes)   
  Welcome letter to Williams Students
Possible thesis / colloquium projects
Opportunities in Mathematics (REUs, gradschool)
photos of Cam and Kayla

RESEARCH INTERESTS:  Analytic Number Theory, Random Matrix Theory, Probability (zeros and n-level statistics for families of L-functions, especially families of elliptic curves with rank over Q(T), classical random matrix theory, random graphs, computational number theory, Benford's Law, cryptography, linear programming, sabermetrics, ...).

      Thesis         Papers        Talks     Books:  Invitation Modern Number Theory           Mathematics of Encryption        Benford's Law       Handouts      My Riddles Page  

ACCEPTED  PAPERS   

  1. 1- and 2-level densities for rational families of elliptic curves: evidence for the underlying group symmetries. Compositio Mathematica (140 (2004), no. 4, 952-992).    pdf
  2. Distribution of eigenvalues for the ensemble of real symmetric Toeplitz matrices (with Chris Hammond). Journal of Theoretical Probability (18 (2005), no. 3, 537-566).   pdf
  3. Benford's law, values of L-functions and the 3x+1 problem (with Alex Kontorovich), Acta Arithmetica(120 (2005), no. 3, 269–297)pdf.
  4. Variation in the number of points on elliptic curves and applications to excess rank, C. R. Math. Rep. Acad. Sci. Canada (27 (2005), no. 4, 111120, Expanded Version). pdf
  5. Incomplete exponential sums in several variables (with Eduardo Dueñez, Howard Straubing, and Amitahba Roy), Journal of Number Theory (116 (2006), no. 1, 168199). pdf
  6. Closed-form Bayesian inferences for the logit model via polynomial expansions (with Eric Bradlow and Kevin Dayaratna), Quantitative Marketing and Economics (4 (2006), no. 2, 173206).   pdf
  7. Investigations of zeros near the central point of elliptic curve L-functions, Experimental Mathematics (15 (2006), no. 3, 257–279).  pdf  (data available online).
  8. The low-lying zeros of a GL(4) and a GL(6) family of L-functions (with Eduardo Dueñez), Compositio Mathematica (142 (2006), no. 6, 1403–1425) pdf.
  9. Low-lying zeros of L-functions with orthogonal symmetry (with Chris Hughes), Duke Mathematical Journal (136 (2007), no. 1, 115–172)   pdf
  10. Constructing one-parameter families of elliptic curves over Q(T) with moderate rank (with Scott Arms, Alvaro Lozano-Robledo), Journal of Number Theory. (123 (2007), no 2, 388–402   pdf
  11. A derivation of James' Pythagorean projection, By The Numbers -- The Newsletter of the SABR Statistical Analysis Committee (16 (February 2006), no. 1, 1722).   pdf (expanded version: pdf).  Chance Magazine (20 (Winter 2007), no. 1, 40-48).
  12. Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and circulant matrices (with Adam Massey and John Sinsheimer), Journal of Theoretical Probability(20 (2007), no. 3, 637--662.)  pdf
  13. Benford's Law applied to hydrology data - results and relevance to other geophysical data (with Mark Nigrini), Mathematical Geology (39 (2007), no. 5, 469--490).  pdf
  14. When the Cramer-Rao Inequality provides no information, Communications in Information and Systems.  (7 (2007), no. 3, 265--272).   pdf
  15. The Modulo 1 Central Limit Theorem and Benford's Law for Products (with Mark Nigrini), International Journal of Algebra. (2 (2008), no. 3, 119--130). pdf
  16. A Symplectic Test of the L-Functions Ratios Conjecture, IMRN (2008). (Vol. 2008, article ID rnm146, 36 pages, doi:10.1093/imrn/rnm146).   pdf
  17. An identity for sums of polylogarithm functions, Integers (electronic journal of combinatorial number theory). (8 (2008), A15)   pdf
  18. A probabilistic proof of Wallis' formula for π, American Mathematical Monthly. (115 (2008), no. 8, 740--745)   pdf
  19. The distribution of the second largest eigenvalue in families of random regular graphs (with Tim Novikoff and Anthony Sabelli), Experimental Mathematics. (17 (2008), no. 2, 231--244.)  pdf
  20. Order statistics and Benford's law (with Mark Nigrini), International Journal of Mathematics and Mathematical Sciences (Volume 2008 (2008), Article ID 382948, 19 pages, doi:10.1155/2008/382948)  pdf
  21. Lower order terms in the 1-level density for families of holomorphic cuspidal newforms, Acta Arithmetica. (137 (2009), 51-98)  pdf
  22. Chains of distributions, hierarchical Bayesian models and Benford's Law (with D. Jang, J. U. Kang, A. Kruckman and J. Kudo), Journal of Algebra, Number Theory: Advances and Applications. (volume 1, number 1 (March 2009), 37--60)   pdf
  23. An orthogonal test of the L-functions Ratios Conjecture, London Mathematical Society. (2009, doi:10.1112/plms/pdp009)   pdf
  24. Silver Scheduler: Demand-Driven Scheduling of Movies in a Multiplex (with Jehoshua Eliashberg, Quintus Hegie, Jason Ho, Dennis Huisman, Sanjeev Swami, Charles B. Weinberg and Berend Wierenga), lead article International Journal of Market Research and won award for best paper of 2009 (doi:10.1016/j.ijresmar.2008.09.004)  pdf
  25. The effect of convolving families of L-functions on the underlying group symmetries (with Eduardo Dueñez), Proceedings of the London Mathematical Society. (2009; doi: 10.1112/plms/pdp018)   pdf
  26. When almost all sets are difference dominated (with Peter Hegarty), Random Structures and Algorithms. (35, 2009, no. 1, 118-136)   pdf 
  27. Nuclei, primes and the random matrix connection (with Frank W. K. Firk), invited paper to Symmetry (1, (2009), 64--105; doi:10.3390/sym1010064)   pdf
  28. Data diagnostics using second order tests of Benford's Law (with Mark Nigrini), Auditing: A Journal of Practice and Theory.  (28 (2009), no. 2, 305--324. doi: 10.2308/aud.2009.28.2.305)  MSWord file
  29. Explicit constructions of infinite families of MSTD sets (with Brooke Orosz and Dan Scheinerman), Journal of Number Theory (130 (2010) 1221--1233) pdf
  30. A unitary test of the L-functions Ratios Conjecture (with John Goes, Steven Jackson, David Montague, Kesinee Ninsuwan, Ryan Peckner and Thuy Pham), Journal of Number Theory. (130 (2010) 2238--2258)  pdf
  31. Towards an `average' version of the Birch and Swinnerton-Dyer Conjecture (with John Goes), Journal of Number Theory (130 (2010), no. 10, 2341--2358)    pdf
  32. An Orthogonal Test of the L-functions Ratios Conjecture, II (with David Montague),  Acta Arithmetica  (146 (2011), 53--90.)  pdf
  33. Rational irrational proofs (with David Montague), Mathematics Magazine (85 (2012), no. 2, 110--114).  pdf (expanded version: pdf;    letter from Nerode and Tennenbaum on history of proof: pdf)
  34. Explicit constructions of infinite families of MSTD sets (with Dan Scheinerman), Additive Number Theory: Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson, David Chudnovsky and Gregory Chudnovsky (Editors), Springer-Verlag, 2010. pdf
  35. Demand-driven scheduling of movies in a multiplex (with Jehoshua Eliashberg and Charles B. Weinberg), newsletter of the European Marketing Academy, October 2010 (requested summary of Silver-Scheduler paper in honor of it receiving the International Journal of Market Research Best Paper Award for 2009).
  36. Effective equidistribution and the Sato-Tate law for families of elliptic curves (with M. Ram Murty), Journal of Number Theory (131 (2011), no. 1, 25-44)   pdf
  37. The lowest eigenvalue in Jacobi ensembles and Painlevé VI (with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina Snaith), Journal of Physics A: Mathematical and Theoretical (43 (2010), 405204, 27pages)   pdf
  38. Isoperimetric Sequences (with Frank Morgan, Edward Newkirk and Lori Pedersen), Mathematics Magazine (84 (Feb. 2011), 37--42).  pdf
  39. Distribution of eigenvalues for highly palindromic real symmetric Toeplitz matrices (with Steven Jackson and Thuy Pham), Journal of Theoretical Probability(25 (2012), 464--495)   pdf
  40. A combinatorial identity for studying Sato-Tate type problems (with M. Ram Murty and F. Strauch), Rendiconti del Seminario Matematico (68 (2010), no. 4, 337--348).  pdf
  41. Moments of the rank of elliptic curves (with Siman Wong), Canadian Journal of Mathematics (64 (2012), no. 1, 151--182)    pdf
  42. The `Real' Schwarz lemma (with David Thompson):  American Mathematical Monthly. (118 (October 2011), no. 8, page 725)   pdf
  43. An elliptic curve family test of the Ratios Conjecture (with Duc Khiem Huynh and Ralph Morrison), Journal of Number Theory (131 (2011), 1117--1147)  pdf
  44. On the number of summands in Zeckendorf decompositions (with Gene S. Kopp, Murat Koloğlu and Yinghui Wang), Fibonacci Quarterly. (49 (2011), no 2, 116--130)   pdf
  45. Generalized More Sums Than Differences Sets (with Geoffrey Iyer, Oleg Lazarev and Liyang Zhang), Journal of Number Theory. (132 (2012), no. 5, 1054--1073) pdf 
  46. Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets (with Luc Robinson and Sean Pegado). INTEGERS (12, #A30 2012). pdf
  47. Models for zeros near the central point of families of elliptic curves (with Eduardo Dueñez, Duc Khiem Huynh, Jon Keating and Nina Snaith), Journal of Physics A: Mathematical and Theoretical (45 (2012),  115207 (32pp)).  pdf
  48. From Fibonacci Numbers to Central Limit Type Theorems (with Yinghui Wang), Journal of Combinatorial Theory, Series A. (7 (2012), 1398-1413)   pdf    (expanded arxiv version)
  49. First Order Approximations of the Pythagorean Won-Loss Formula for Predicting MLB Teams Winning Percentages (with Kevin Dayaratna), By The Numbers -- The Newsletter of the SABR Statistical Analysis Committee,   pdf (expanded version with appendix proving main result with just one variable calculus: pdf); entire issue of By The Numbers here (22 (2012), no 1, 15--19).
  50. Low-lying zeros of number field L-functions (with Ryan Peckner), Journal of Number Theory.  (132 (2012), 2866--2891) pdf
  51. Quadratic fields with cyclic 2-class groups (with Carlos Dominguez and Siman Wong): Journal of Number Theory (133 (2013), no. 3, 926--939).   pdf
  52. Leading, learning and herding (with Daniel Stone), Mathematical Social Sciences (65 (2013), 222--231)  pdf
  53. The Pythagorean Won-Loss Formula and Hockey: A Statistical Justification for Using the Classic Baseball Formula as an Evaluative Tool in Hockey (with Kevin Dayaratna), The Hockey Research Journal: A Publication of the Society for International Hockey Research. ((2012/2013), pages 193--209)   pdf
  54. The Average Gap Distribution for Generalized Zeckendorf Decompositions (with Olivia Beckwith, Amanda Bower, Louis Gaudet, Rachel Insoft, Shiyu Li and Philip Tosteson), the Fibonacci Quarterly (51 (2013), 13--27). pdf
  55. Closed Form Continued Fraction Expansions of Special Quadratic Irrationals (with Dan Fishman), ISRN Combinatorics. (vol. 2013, Article ID 414623, 5 pages)  pdf (journal version)
  56. Distribution of Missing Sums in Sumsets (with Oleg Lazarev and Kevin O'Bryant), Experimental Mathematics (22 (2013), no. 2, 132--156).  pdf
  57. The n-level densities of low-lying zeros of quadratic Dirichlet L-function (with Jake Levinson), Acta Arithmetica. (161 (2013), 145--182)   pdf   (Mathematica files available here)
  58. Viruslike dynamics on starlike graphsviruslike dynamics on starlike graphs (with Thealexa Becker, Alec Greaves-Tunnell, Leo Kontorovich, Amitabha Roy, Karen Shen), the Journal of Nonlinear Systems and Applications (4 2013), no. 153--63). pdf
  59. The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices (with Gene S. Kopp Murat Koloğlu, Frederick Strauch, Wentao Xiong). Journal of Theoretical Probability  (26 (2013), no. 4, 1020--1060) pdf
  60. Coordinate sum and difference sets of d-dimensional modular hyperbolas (with Amanda Bower, Ron Evans and Victor Luo),  INTEGERS (#A31, 2013, 16 pages)  pdf  (expanded arxiv version: pdf)
  61. The Pi Mu Epsilon 100th Anniversary Problems: Part I (with James M. Andrews, Avery T. Carr and many of my students), The Pi Mu Epsilon Journal (13 (2013), no. 9, 513--534). pdf
  62. The low-lying zeros of level 1 Maass forms (with Levent Alpoge), IMRN. (2014, 24 pages, doi:10.1093/imrn/rnu012) pdf
  63. Special Sets of Primes in Function Fields (with Julio Andrade, Kyle Pratt and Minh-Tam Trinh), INTEGERS. (14 (2014), #A18) pdf 
  64. Generalizing Zeckendorf's Theorem to \(f\)-decompositions (joint with Philippe Demontigny, Thao Do, Archit Kulkarni, David Moon and Umang Varma), Journal of Number Theory. (141 (2014), 136--158)  pdf
  65. Pythagoras at the Bat (with Taylor Corcoran, Jen Gossels, Victor Luo and Jaclyn Porfilio), book chapter in Social Networks and the Economics of Sports (edited by Panos M. Pardalos and Victor Zamaraev, Springer-Verlag, 2014, pages 89--114).  pdf
  66. A Generalization of Fibonacci Far-Difference Representations and Gaussian Behavior (with Philippe Demontigny, Thao Do, Archit Kulkarni and Umang Varma), Fibonacci Quarterly (52 (2014), no. 3, 247--273)  pdf
  67. The Pi Mu Epsilon 100th Anniversary Problems: Part II (with James M. Andrews, Avery T. Carr and many of my students), The Pi Mu Epsilon Journal (14 (2013), no. 10, 577--608). pdf
  68. Newman's conjecture in various settings (with Julio Andrade and Alan Chang),  Journal of Number Theory. (144 (2014), 70--91pdf
  69. Maass waveforms and low-lying zeroes (with Levent Alpoge, Nadine Amersi, Geoffrey Iyer, Oleg Lazarev and Liyang Zhang), ``Analytic Number Theory: In honor of Helmut Maier's 60th birthday'' (Carl Pomerance, Michael Th. Rassias, editors), Springer-Verlag 2015.  pdf
  70. Generalized Ramanujan Primes (with Nadine Amersi, Olivia Beckwith, Ryan Ronan and Jonathan Sondow). Combinatorial and Additive Number Theory:  CANT 2011 and 2012 (Springer Proceedings in Mathematics & Statistics, Melvyn B. Nathanson editor, 2014, pages 1--13). pdf
  71. Finding and Counting MSTD sets (with Geoffrey Iyer, Oleg Lazarev and Liyang Zhang). Combinatorial and Additive Number Theory:  CANT 2011 and 2012 (Springer Proceedings in Mathematics & Statistics, Melvyn B. Nathanson editor, 2014, pages 79--98).  pdf
  72. Most Subsets are Balanced in Finite Groups (with Kevin Vissuet), Combinatorial and Additive Number Theory:  CANT 2011 and 2012 (Springer Proceedings in Mathematics & Statistics, Melvyn B. Nathanson editor, 2014, pages 147--157)  pdf
  73. Gaussian Behavior in Generalized Zeckendorf Decompositions (with Yinghui Wang), Combinatorial and Additive Number Theory:  CANT 2011 and 2012 (Springer Proceedings in Mathematics & Statistics, Melvyn B. Nathanson editor, 2014, pages 159--173).   pdf
  74. The Pi Mu Epsilon 100th Anniversary Problems: Part III (with many of my students), The Pi Mu Epsilon Journal (14fc (2014), no. 1, 65--99). pdf  
  75. Continued Fraction Digit Averages and Maclaurin's Inequalities (with Francesco Cellarosi, Doug Hensley and Jake Wellens), Experimental Mathematics (24 (2015), no. 1, 23--44)  pdf
  76. The \(n\)-Level Density of Dirichlet \(L\)-Functions over \(\mathbb{F}_q[T]\) (with Julio Andrade, Kyle Pratt and Minh-Tam Trinh), Communications in Number Theory and Physics. (8 (2014), no. 3, 511-539)  pdf
  77. The expected eigenvalue distribution of large, weighted \(d\)-regular graphs (with Leo Goldmakher, Cap Khoury and Kesinee Ninsuwan). Random Matrices: Theory and Applications. (3 (2014), no. 2, 1450015, 22 pages)  pdf
  78. The Pi Mu Epsilon 100th Anniversary Problems: Part IV (with many of my students), The Pi Mu Epsilon Journal (15 (2015), no. 2, 100--134) pdf
  79. The James Function (with Christopher N. B. Hammond and Warren P. Johnson),  Mathematics Magazine. (88 (2015), 54--71). pdf
  80. Benford Behavior of Zeckendorf Decompositions (with A. Best, P. Dynes, X. Edelsbrunner, B. McDonald, C. Turnage-Butterbaugh and M. Weinstein),  Fibonacci Quarterly. (52 (2014), no. 5, 35--46)  pdf
  81. Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals (with A. Best, P. Dynes, X. Edelsbrunner, B. McDonald C. Turnage-Butterbaugh and M. Weinstein),  Fibonacci Quarterly. (52 (2014), no. 5, 47--53) pdf
  82. Distribution of eigenvalues of weighted, structured matrix ensembles (with Olivia Beckwith, Victor Luo, Karen Shen and Nicholas Triantafillou), INTEGERS(15 (2015), paper A21, 28 pages) pdf 
  83. Limiting Spectral Measures for Random Matrix Ensembles with a Polynomial Link Function (with Kirk Swanson, Kimsy Tor and Karl Winsor), Random Matrices: Theory and Applications (4 (2015), no. 2, 1550004, 28 pages) pdf
  84. Relieving and Readjusting Pythagoras  (with Victor Luo), By The Numbers -- The Newsletter of the SABR Statistical Analysis Committee. (25 (2015), no. 1, 5-14) pdf  (older, expanded version: pdf)
  85. Sums and Differences of Correlated Random Sets (with Thao Do, Archit Kulkarni, David Moon and Jake Wellens) Journal of Number Theory(147 (2015), 44--68) pdf
  86. Generalizing Zeckendorf's Theorem: The Kentucky Sequence (with M. Catral, P. Ford, P. Harris and D. Nelson), Fibonacci Quarterly. (52 (2014), no. 5, 68--90)  pdf
  87. Gaps between zeros of GL(2) L-functions (with Owen Barrett, Brian McDonald, Ryan Patrick, Caroline Turnage-Butterbaugh and Karl Winsor), Journal of Mathematical Analysis and Applications.  (429 (2015), 204--232) pdf
  88. Surpassing the Ratios Conjecture in the 1-level density of Dirichlet L-functions (with Daniel Fiorilli), Algebra & Number Theory  (9 (2015), no. 1, 13--52) pdf (expanded arXiv version)
  89. Gaps between summands in generalized Zeckendorf decompositions  (with Amanda Bower, Rachel Insoft, Shiyu Li and Philip Tosteson, and an appendix with Iddo Ben-Ari), Journal of Combinatorial Theory, Series A(135 (2015), 130--160) pdf
  90. Sets Characterized by Missing Sums and Differences in Dilating Polytopes (with Thao Do, Archit Kulkarni, David Moon and Jake Wellens), Journal of Number Theory. (157 (2015), 123--153) pdf
  91. Newman's Conjecture in Function Fields (with Alan Chang, David Mehrle, Tomer Reiter, Joseph Stahl and Dylan Yott), Journal of Number Theory. (15 (2015), 154--169) pdf
  92. Equipartitions and a Distribution for Numbers: A Statistical Model for Benford's Law (with Joe Iafrate and Frederick Strauch), Physical Review E (91 (2015), no. 6, 062138 (6 pages))   pdf
  93. Lower-Order Biases in Elliptic Curve Fourier Coefficients in Families (with Blake Mackall, Christina Rapti and Karl Winsor), Frobenius Distributions: Lang-Trotter and Sato-Tate Conjectures (David Kohel and Igor Shparlinski, editors), Contemporary Mathematics 663, AMS, Providence, RI 2016. pdf
  94. From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond (with Owen Barrett, Frank W. K. Firk and Caroline Turnage-Butterbaugh), Open Problems in Mathematics (editors John Nash  Jr. and Michael Th. Rassias, Springer-Verlag, 2016). pdf
  95. Determining Optimal Test Functions for Bounding the Average Rank in Families of L-Functions} (with Jesse Freeman), in SCHOLAR -- a Scientific Celebration Highlighting Open Lines of Arithmetic Research, Conference in Honour of M. Ram Murty's Mathematical Legacy on his 60th Birthday (A. C. Cojocaru, C. David and F. Pappaardi, editors), Contemporary Mathematics 655, AMS and CRM, 2015.. pdf
  96. The Weibull distribution and Benford's Law (with Victoria Cuff and Allie Lewis), Involve (8-5 (2015), 859--874. DOI 10.2140/involve.2015.8.859).  pdf
  97. Some Results in the Theory of Low-lying Zeros (with Blake Mackall, Christina Rapti, Caroline Turnage-Butterbaugh and Karl Winsor, and an appendix with Megumi Asada, Eva Fourakis, Kevin Yang), in Families of automorphic forms and the trace formula (editors), Simons Symposia series, Springer-Verlag. pdf
  98. Leading Digit Laws on Linear Lie Groups (with Corey Manack), Research in Number Theory (1:22  (2015), DOI 10.1007/s40993-015-0024-4).  pdf
  99. Gaussian Distribution of the Number of Summands in Generalized Zeckendorf Decompositions in Small Intervals  (with A. Best, P. Dynes, X. Edelsbrunner, B. McDonald C. Turnage-Butterbaugh and M. Weinstein), INTEGERS (16 (2016), #A6) pdf
  100. Crescent configurations (with David Burt, Eli Goldstein, Sarah Manski, Eyvindur Ari Palsson and Hong Suh), Integers (electronic journal of combinatorial number theory) (16 (2016), #A38)    pdf
  101. Multiple Regression Analysis: Understanding the Impact of Offensive and Defensive Contributions to Team Performance (with Kevin D. Dayaratna),  The Hockey Research Journal (2014/2015, 41--43).   pdf
  102. A Probabilistic Approach to Generalized Zeckendorf Decompositions (with Iddo Ben-Ari), SIAM Journal on Discrete Mathematics (30 (2016), no. 2, 1302--1332) pdf  (journal version here)
  103. Individual Gap Measures from Generalized Zeckendorf Decompositions (with Robert Dorward, Pari Ford, Eva Fourakis, Pamela Harris, Eyvindur Palsson, and Hannah Paugh), Uniform Distribution Theory (12,(2017), no. 1, 27--36). pdf
  104. A Generalization of Zeckendorf's Theorem via Circumscribed m-gons (with Robert Dorward, Pari L. Ford, Eva Fourakis, Pamela E. Harris, Eyvindur A. Palsson and Hannah Paugh), to appear in Involve. pdf
  105. Geometric-Progression-Free Sets over Quadratic Number Fields (with Andrew Best, Karen Huan, Nathan McNew, Jasmine Powell, Kimsy Tor and Madeleine Weinstein), Proceedings of the Royal Society of Edinburgh, Section A: Mathematics (147 (2017), no. 2, 242-262).  pdf
  106. The M&M Game: From Morsels to Modern Mathematics (with Ivan Badinski, Nathan McCue, Cameron Miller, Kayla Miller and Michael Stone). Mathematics Magazine (90 (2017), no. 3, 197--207)  pdf (pruned published version: pdf: Copyright [2017] Mathematical Association of America. All Rights Reserved.).
  107. Legal Decompositions Arising from Non-positive Linear Recurrences (with Minerva Catral, Pari Ford, Pamela Harris, Steven J. Miller, Dawn Nelson), Fibonacci Quarterly  (54 (2016), no. 4, 348--365).  pdf (journal version: pdf)
  108. Subsets of \(\mathbb{F}_q[x]\) Free of 3-term Geometric Progressions (with Megumi Asada, Eva Fourakis, Sarah Manski, Gwyneth Moreland and Nathan McNew), to appear in Finite Fields and their Applicationspdf
  109. Fringe pairs in generalized MSTD sets (with Megumi Asada, Sarah Manski and Hong Suh), International Journal of Number Theory. (13 (2017), no. 2, 2653--2675) pdf
  110. On Identities of Ruggles, Horadam, Howard, And Young (with Curtis Cooper, Peter J. C. Moses, Murat Sahin, and Thotsaporn Thanatipanonda), the Proceedings of the 17th International Fibonacci Conference, Fibonacci Quarterly (55 (2017), no. 5, 52--65). pdf
  111. A Collection of Central Limit Type Results in Generalized Zeckendorf Decompositions (with Ray Li), Proceedings of the Seventeenth International Conference on Fibonacci Numbers and Their Applications,
    Fibonacci Quarterly (55 (2017), no. 5, 105--114).  pdf
  112. On the Asymptotic Behavior of Variance of PLRS Decompositions (with Dawn Nelson, Zhao Pan and Huanzhong Xu), the Proceedings of the 17th International Fibonacci Conference, Fibonacci Quarterly (55 (2017), no. 5, 135--143).  pdf
  113. If a prime divides a product (with Cesar Silva), College Mathematics Journal. (48 (2017), no. 2, 123--128)   pdf
  114. New Behavior in Legal Decompositions Arising from Non-positive Linear Recurrences (with Minerva Catral, Pari Ford, Pamela Harris, Steven J. Miller, Dawn Nelson, Zhao Pan and Huanzhong Xu), Fibonacci Quarterly (55 (2017), no. 3, 252--275). 
  115. The Emergence of 4-Cycles in Polynomial Maps over the Extended Integers (with Andrew Best, Patrick Dynes, Jasmine Powell and Ben Weiss, Minnesota Journal of Undergraduate Mathematics(2 (2016-2017), no. 1, 15 pages) pdf
  116. Low-lying zeros for L-functions associated to Hilbert modular forms of large level) (with Sheng-Chi Liu), Acta Arithmetica (180 (2017), 251--266 pdf
  117. Constructing families of moderate-rank elliptic curves over number fields (with David Mehrle, Tomer Reiter, Joseph Stahl and Dylan Yott), Minnesota Journal of Undergraduate Mathematics (2 (2016-2017), 11 pages)  pdf
  118. Benford Behavior of Generalized Zeckendorf Decompositions  (with A. Best, P. Dynes, X. Edelsbrunner, B. McDonald C. Turnage-Butterbaugh and M. Weinstein), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer, New York, 2017. pdf
  119. Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions (with Andrew Best, Karen Huan, Nathan Mcnew, Steven J. Miller, Jasmine Powell, Kimsy Tor and Madeleine Weinstein), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer, New York, 2017. pdf
  120. One-Level density for holomorphic cusp forms of arbitrary level (with Owen Barrett, Paula Burkhardt, Robert Dorward and Jon DeWitt), to appear in Research in Number Theory  pdf
  121. Dimensional lower bounds for Falconer type incidence and point configuration theorems (with Jonathan DeWitt, Kevin Ford, Eli Goldstein, Gwyneth Moreland, Eyvindur A. Palsson, Steven Senger), to appear in the Journal d'Analyse Mathematique.   pdf
  122. Irrationality measure and lower bounds for \(\pi(x)\) (with David Burt, Sam Donow, Matthew Schiffman and Ben Wieland), to appear in the The Pi Mu Epsilon Journal. pdf
  123. When almost all sets are difference dominated in \(\mathbb{Z}/n\mathbb{Z}\) (with Anand Hemmady and Adam Lott), to appear in INTEGERS. pdf
  124. On smoothing singularities of elliptic orbital integrals on GL(n) and Beyond Endoscopy (with Oscar Gonzalez, Chung Hang Kwan, Roger Van Peski and Tian An Wong), to appear in the Journal of Number Theory.    pdf
  125. Optimal point sets determining few distinct triangles (with Alyssa Epstein, Adam Lott and Eyvindur Palsson), to appear in INTEGERS.   pdf
  126. Benford's Law and Continuous Dependent Random Variables (with Thealexa Becker, David Burt, Taylor C. Corcoran, Alec Greaves-Tunnell, Joseph R. Iafrate, Joy Jing, Jaclyn D. Porfilio, Ryan Ronan, Jirapat Samranvedhya, Blaine Talbut and Frederick W. Strauch), to appear in the Annals of Physics. pdf.
  127. Problem Session Notes from CANT Conferences (currently 2009 through 2016, inclusive); distributed to interested parties and posted on the arXiv, 35 pages. pdf

SUBMITTED  PAPERS   

  1. When Sets Can and Cannot Have MSTD Subsets (with Nathan McNew, Victor Xu and Sean Zhang). submitted to INTEGERS. pdf
  2. On Summand Minimality of Generalized Zeckendorf Decompositions (with Katherine Cordwell, Max Hlavacek, Chi Huynh, Carsten Peterson, and Yen Nhi Truong Vu), Research in Number Theory pdf
  3. Central Limit Theorems for Gaps of Generalized Zeckendorf Decompositions (with Ray Li), submitted to the Fibonacci Quarterlypdf
  4. Random Matrix Ensembles with Split Limiting Behavior (with Paula Burkhardt, Peter Cohen, Jonathan Dewitt, Max Hlavacek, Carsten Sprunger, Yen Nhi Truong Vu, Roger Van Peski, and Kevin Yang, and an appendix joint with  Manuel Fernandez and Nicholas Sieger), submitted to  Random Matrices: Theory and Applications  pdf
  5. The Inverse Gamma Distribution and Benford's Law (with Rebecca F. Durst, Chi Huynh, Adam Lott, Eyvindur A. Palsson, Wouter Touw, and Gert Vriend), submitted to Minnesota Journal of Undergraduate Mathematics  pdf
  6. Classification of crescent configurations (with Rebecca F. Durst, Max Hlavacek, Chi Huynh, Steven J. Miller and Eyvindur A. Palsson), submitted to INTEGERS.   pdf
  7. On Within-Perfectness and Near-Perfectness (with Peter Cohen, Katherine Cordwell, Alyssa Epstein, Chung Hang Kwan and Adam Lott) submitted to Acta Arithmetica.   pdf
  8. On algorithms to calculate integer complexity (with Katherine Cordwell, Alyssa Epstein, Anand Hemmady, Eyvindur A. Palsson, Aaditya Sharma, Stefan Steinerberger, Yen Nhi Truong Vu), submitted to INTEGERS. pdf
  9. The bidirectional ballot polytope (with Carsten Peterson, Carsten Sprunger and Roger Van Peski), submitted to INTEGERS. pdf
  10. A geometric perspective on the MSTD question (with Carsten Peterson). Submitted to Discrete and Computational Geometrypdf

PREPRINTS / IN  PREPARATION (available upon request)   

  1. Avoiding 3-Term Geometric Progressions in Non-Commutative Settings (with Megumi Asada, Eva Fourakis, Eli Goldstein, Sarah Manski, Nathan Mcnew and Gwyneth Moreland), pdf
  2. The Ratios Conjecture and Convolving Families of \(L\)-functions (with Minh-Tam Trinh).
  3. When Generalized Sumsets are Difference Dominated (with Virginia Hogan). pdf
  4. Moment Formulas for Ensembles of Classical Compact Groups (with Geoffrey Iyer and Nicholas Triantafillou)
  5. \(m\)-Paths and the \(3x+1\) Problem (with Bruce Adcock, Carlos Dominguez and Sucheta Soundararajan).

BOOKS   

  1. An Invitation to Modern Number Theory (with Ramin Takloo-Bighash), Princeton University Press, 2006, 519 pages.
  2. The Mathematics of Encryption: An Elementary Introduction (with Midge Cozzens), AMS Mathematical World series 29, Providence, RI, 2013, 332 pages.
  3. Theory and Applications of Benford's law (senior editor for book, duties include writing chapters and coordinating contributions): Princeton University Press, 2015, 438 pages.
  4. The Probability Lifesaver: Princeton University Press, 2017, 752 pages.
  5. Mathematics of Optimization: How to do things faster: AMS, to appear.
  6. The Pi Mu Epsilon 100th Anniversary Problems (with Stephan Garcia): In preparation.

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RECENT CONFERENCES:  Undergrad Research Panel (Boston 2012)      Undergrad Research Session (HC 2011 and BC 2013)