FIBONACCI

• Zeckendorf Decompositions (existance, Gaussianity)

  • 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
  • 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)
  • 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
  • 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
  • Gausssian 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
  • 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
  • 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
  • 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)
  • 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
  • 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
  • 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). pdf
  • 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  (4 (2018), no. 43, https://doi.org/10.1007/s40993-018-0137-7) pdf
  • Limiting Distributions in Generalized Zeckendorf Decompositions (with Alex Gueganic, Granger Carty, Yujin H. Kim, Alina Shubina, Shannon Sweitzer, Eric Winsor and Jianing Yang), Fibonacci Quarterly. (57 (2019), no. 2, 109--125) pdf
  • Gaussian Behavior in Zeckendorf Decompositions From Lattices (with Eric Chen, Robin Chen, Lucy Guo, Cindy Jiang, Joshua M. Siktar, Peter Yu), Fibonacci Quarterly. (57 (2019), no. 3, 201--212) pdf
  • Central Limit Theorems for Gaps of Generalized Zeckendorf Decompositions (with Ray Li), Fibonacci Quarterly. (57 (2019), no. 3, 213--230)  pdf
  • Zeckendorf's Theorem Using Indices in an Arithmetic Progression (with Amelia Gilson, Hadley Killen, Tamas Lengyel, Nadia Razek, Joshua M. Siktar, and Liza Sulkin,  Fibonacci Quarterly.  (59 (2021), 327--337) pdf
  • On Zeckendorf related partitions using the Lucas sequence (with Hung Viet Chu and David Luo), Fibonacci Quarterly.  (60 (2022), 111--119) pdf (arxiv version: pdf)
  • Generalizing Zeckendorf's Theorem to Homogeneous Linear Recurrences, I (with Thomas Martinez, Clay Mizgerd and Chenyang Sun), Fibonacci Quarterly.  (Conference Proceedings: 20th International Fibonacci Conference: 60 (2022), no.5, 222--230). pdf
  • Generalizing Zeckendorf's Theorem to Homogeneous Linear Recurrences, II (with Thomas Martinez, Clay Mizgerd, Jack Murphy and Chenyang Sun), Fibonacci Quarterly.  (Conference Proceedings: 20th International Fibonacci Conference: 60 (2022), no.5, 231--254). pdf

• Zeckendorf Decompositions (gaps among summands)

  • 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
  • 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
  • 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
  • Gaps of Summands of the Zeckendorf Lattice (with Neelima Borade, Dexter Cai, David Z. Chang, Bruce Fang, Alex Liang and Wanqiao Xu), Fibonacci Quarterly. (2 (2020), 143-156),  pdf

• Zeckendorf Games

  • The Zeckendorf Game (with Paul Baird-Smith, Alyssa Epstein and Kristen Flint), to appear in the Proceedings of CANT 2018.  pdf
  • The Generalized Zeckendorf Game (with Paul Baird-Smith, Alyssa Epstein and Kristen Flint), Fibonacci Quarterly (57 (2019) no. 5, 1-14)  pdf
  • Extending Zeckendorf's Theorem to a Non-constant Recurrence and the Zeckendorf Game on this Non-constant Recurrence Relation (with Ela Boldyriew, Anna Cusenza, Linglong Dai, Pei Ding, Aidan Dunkelberg, John Haviland, Kate Huffman, Dianhui Ke, Daniel Kleber, Jason Kuretski, John Lentfer, Tianhao Luo, Clayton Mizgerd, Vashisth Tiwari, Jingkai Ye, Yunhao Zhang, Xiaoyan Zheng, and Weiduo Zhu), Fibonacci Quarterly. (5 (2020), 55-76)  pdf
  • Deterministic Zeckendorf Games (with Ruoci Li, Xiaonan Li, Clay Mizgerd, Chenyang Sun, Dong Xia, And Zhyi Zhou), Fibonacci Quarterly. (58 (2020), no. 5, 152-160)  p3df
  • Winning Strategy for the Multiplayer and Multialliance Zeckendorf Games (with Anna Cusenza, Aiden Dunkelberg, Kate Huffman, Dianhui Ke, Daniel Kleber, Clayton Mizgerd, Vashisth Tiwari, Jingkai Ye, Xiaoyan Zheng), Fibonacci Quarterly. (59 (2021), 308--318)  pdf
  • Bounds on Zeckendorf Games (with Anna Cusenza, Aiden Dunkelberg, Kate Huffman, Dianhui Ke, Daniel Kleber, Micah McClatchey, Clayton Mizgerd, Vashisth Tiwari, Jingkai Ye, Xiaoyan Zheng), Fibonacci Quarterly. pdf  (1 (2022), no. 1, 57--71)   
  • Winning Strategies for the Generalized Zeckendorf Game (Steven J. Miller, Eliel Sosis, Jingkai Ye), Fibonacci Quarterly.  (Conference Proceedings: 20th International Fibonacci Conference: 60 (2022), no.5, 270--292). pdf
  • The Accelerated Zeckendorf Game (Diego Garcia-Fernandezsesma, Steven J. Miller, Thomas Rascon, Risa Vandegrift, and Ajmain Yamin), to appear in the Fibonacci Quarterly.  pdf
  • Towards the Gaussianity of Random Zeckendorf Games (Justin Cheigh, Guilherme Zeus Dantas E Moura, Ryan Jeong, Jacob Lehmann Duke, Wyatt Milgrim, Steven J. Miller, and Prakod Ngamlamai), submitted to the CANT Proceedings.    pdf
• Zeckendorf and Benford's Law
  • 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

  • Benford's Law under Zeckendorf Decomposition (Sungkon Chang, Steven J. Miller).    pdf

  • 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

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MY BOOKS

• Invitation to Modern Number Theory

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• Benford's Law

• Probability Lifesaver  

• Mathematics of Optimization

• Pi Mu Epsilon Centennial Collection