Research

Research articles published:


[48]  [pdf] Sinai Robins,  The integer point transform as a complete invariant, Communications in Mathematics, 31 (2023), no. 2, 157-172. 

DOI: https://doi.org/10.46298/cm.11218

[47]  [pdf] Fabricio Caluza Machado and Sinai Robins,  The null set of a polytope and the Pompeiu property for polytopes, Journal dÁnalyse Mathématique,  150 (2023), 673-683.

[46]  [pdf]  Luca Brandolini, Leonardo Colzani, Sinai Robins, and Giancarlo Travaglini,   An Euler-MacLaurin formula for polygonal sums, Transactions of the AMS, 375 (2022), 151-172. 

[45]  [pdf] C. G. Fernandes, J. C. de Pina, J. L. Ramirez Alfonsin, S. Robins, Period collapse in Ehrhart quasi-polynomials of graphs, Combinatorial Theory,  2(3), (2022), 1-43. 

[44]  [pdf]  Cristina G. Fernandes,  Jose Coelho de Pina,  Jorge Luis Ramirez Alfonsin, and Sinai Robins,  Cubic graphs, their Ehrhart quasi-polynomials, and a scissors congruence phenomenon,  Discrete and Computational Geometry, Vol. 65, (2021), no. 1, 227-243.  

[43]  [pdf]  Luca Brandolini, Leonardo Colzani, Sinai Robins, and Giancarlo Travaglini, Pick's Theorem and Convergence of multiple Fourier series, The American Mathematical Monthly, (2021),  41-49.

[42]  [pdf]   Imre Barany, Greg Martin, Eric Naslund, and Sinai Robins,  Primitive points in lattice polygons,  The Canadian Mathematical Bulletin, Vol. 63, (2020), no. 4, 850-870. 

[41]  [pdf]  Alexander Kolpakov and Sinai Robins, Spherical tetrahedra with rational volume, and spherical Pythagorean triples, Math. Comp. Vol. 89, (2020), no. 324, 2031-2046. 

DOI: https://doi.org/10.1090/mcom/3496

[40]  [pdf]   Le, Quang-Nhat,  Sinai Robins,  Christophe Vignat,  Tanay Wakhare,  A continuous analogue of lattice path enumeration, Electronic J.  Combinatorics 26, issue 3, (2019), 1-13.

[39]  [pdf]  C. C. Haessig, A. Iosevich, J. Pakianathan, S. Robins, and L. Vaicunas, Tilings, circle packings, and exponential sums over finite fields, Analysis Mathematica, 44, (2018), 433-449.          

[38]  [pdf]  Imre Barany, Arseniy Akopyan, and Sinai Robins, Algebraic vertices of non-convex polyhedra, Advances in Math, 308, (2017), 627-644.

[37]  [pdf]  Maciej Borodzik, Danny Nguyen, and Sinai Robins, Tiling the integer lattice with translated sublattices, Moscow Journal of Combinatorics and Number Theory, Vol 6, issue 4, 2016, 3-26.

[36]  [pdf]  Romanos-Diogenes Malikiosis, Sinai Robins, and Zhang Yichi, Polyhedral Gauss sums, and polytopes with symmetry,  Journal of Computational Geometry, Vol 7, no. 1, 2016, 149-170.

[35]  [pdf]  Amanda Folsom, Winfried Kohnen, and Sinai Robins, Cone theta functions and spherical polytopes with rational volumes,  Annales de L’Institut Fourier, Vol 65, no. 3, 2015, 1133-1151.

[34]  [pdf]  Matthias Beck and Sinai Robins, Computing the continuous discretely. Integer-point enumeration in polyhedra, Second edition, with illustrations by David Austin, Undergraduate Texts in Mathematics. Springer, New York, 2015, xx+285 pp.   ISBN: 978-1-4939-2968-9; 978-1-4939-2969-6

[33]  [pdf]  Sergei Tabachnikov, Pierre Deligne, and Sinai Robins, The Ice Cube Proof, The Mathematical Intelligencer, Vol 36, no. 4, 2014, 1-3.   

[32]  [pdf]  Karl Dilcher and Sinai Robins, Zeros and irreducibility of polynomials with GCD powers as coefficients, The Ramanujan Journal, Vol. 36, Issue 1-2, Feb. 2015, 227-236.  

[31]  [pdf]  Nick Gravin, Fedor Petrov, Sinai Robins, and Dmitry Shiryaev, Convex curves and a Poisson imitation of lattices, Mathematika (January 2014), Vol 60, no. 1, 139-152.  

[30]  [pdf]  Nick Gravin, Mihail Kolountzakis, Sinai Robins, and Dmitry Shiryaev, Structure results for multiple tilings in 3D, Discrete & Computational Geometry, (2013), Vol. 50, 1033-1050.  

[29]  [pdf]  Stanislav Jabuka, Sinai Robins, and Xinli Wang, Heegaard Floer correction terms and Dedekind-Rademacher sums, Int. Math. Res. Notices IMRN (2013), no. 1, 170-183. 

[28]  [pdf]  Nick Gravin, Sinai Robins, and Dima Shiryaev, Translational tilings by a polytope, with multiplicity, Combinatorica, 32 (6),  (2012) 629-648.   

[27]  [pdf]  Nick Gravin, Jean Lasserre, Dmitrii Pasechnik, and Sinai Robins, The inverse moment problem for convex polytopes, Discrete & Computational Geometry, Vol. 48, Issue 3, (2012), 596-621. 

[26]  [pdf]  Stanislav Jabuka, Sinai Robins, and Xinli Wang, When are two Dedekind sums equal?, International Journal of Number Theory, Vol. 7, (8), (2011), 2197-2202.     

[25]  [pdf]  David DeSario and Sinai Robins, Generalized solid angle theory for real polytopes, The Quarterly Journal of Mathematics, Vol. 62, 4, (2011), 1003-1015.  

[24]  [pdf]  David Feldman, Jim Propp, and Sinai Robins, Tiling lattices with sublattices I, Discrete & Computational Geometry, Vol. 46, No. 1, (2011), 184-186.  

[23]  [pdf]  Lenny Fukshansky and Sinai Robins, Bounds for solid angles of lattices of rank 3, Journal of Combinatorial Theory, Series A, 118 (2), (2011), 690-701.  

[22]  [pdf]  Victor Moll, Sinai Robins, and Kirk Soodhalter, The action of Hecke operators on hypergeometric functions, Journal of the Australian Mathematical Society, 89, (2010), 51-74.   

[21]  [pdf]  Matthias Beck, Sinai Robins, and Steven Sam, Positivity theorems for solid-angle polynomials, Beitrage zur Algebra und Geometrie, Vol. 51, No. 2, (2010), 493-507. 

[20]   [pdf]  Matthias Beck and Sinai Robins, Computing the continuous discretely: integer-point enumeration in polyhedra. Springer Undergraduate Texts in Mathematics, 226 pages, (2007). 

[19]  [pdf]   Lenny Fukshansky and Sinai Robins, The Frobenius problem and the covering radius of a lattice,  Discrete & Computational Geometry, 37 (2007), 471–483.  

[18]  [pdf]   Matthias Beck, Sinai Robins, and Shelly Zacks, Higher-dimensional Dedekind sums and their bounds arising from the discrete diagonal of the n-cube, Advances in Applied Mathematics 36, no. 1 (2006), 1–29.  

[17]  [pdf]  Juan Gil and Sinai Robins, Hecke operators on rational functions I, Forum Math. 17 (2005), no. 4, 519–554.  

[16]  [pdf]  Matthias Beck and Sinai Robins, Dedekind sums: a combinatorial-geometric viewpoint, Unusual Applications of Number Theory (M. Nathanson, ed.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science 64 (2004), 25–35. Contemporary Mathematics 374 (2005), 15–36.   

[15]  [pdf]  Matthias Beck and Sinai Robins, A formula related to the Frobenius problem in two dimensions, Number Theory. New York Seminar 2003 (D. Chudnovsky, G. Chudnovsky, M. Nathanson, eds.), pp. 17–23. Springer, Berlin, (2004). 

[14]  [pdf]  Problems from the Cottonwood Room (with Matthias Beck, Beifang Chen, Christian Haase, Allen Knutson, Bruce Reznick, and Achill Schuermann) , Contemporary Mathematics 374 (2005), 179-191 (Proceedings of the Summer AMS/MAA/SIAM Research Conference on Integer Points in Polyhedra, July 13 - July 17, 2003 in Snowbird, Utah)   

[13]  [pdf]  Matthias Beck and Sinai Robins, Explicit and efficient formulas for the lattice point count inside rational polygons, Discrete & Computational Geometry 27 (2002), 443–459.  

[12]  [pdf]  Matthias Beck, Ricardo Diaz, and Sinai Robins, The Frobenius problem, rational polytopes, and Fourier-Dedekind sums, Journal of Number Theory 96 (2002), 1–21.  

[11]  [pdf]  Marvin Knopp and Sinai Robins, Simple proofs of Riemann's functional equation and of the Lipschitz summation formula, Proceedings of the AMS, 129, (2001), No. 7, 1915-1922. 

[10]  [pdf]  Wayne Haga and Sinai Robins, On Kruskals Principle, Organic Mathematics (Burnaby, BC, 1995), 407-412, Canadian Math Society Conf. Proc., 20, Amer. Math. Soc., Providence, RI, (1998).  

[9]  [pdf]  Ricardo Diaz and Sinai Robins, The Ehrhart Polynomial of a Lattice Polytope, Annals of Mathematics, 145, (1997), 503-518.  

[8]  Ricardo Diaz and Sinai Robins, The Ehrhart Polynomial of a Lattice Polytope, Erratum, Annals of Mathematics, 2, 146, (1997), No. 1, 237.  (Note:  this is simply some typos found in the example at the end of paper [9])

[7]  [pdf]  Ricardo Diaz and Sinai Robins, The Ehrhart Polynomial of a Lattice n-simplex, Electronic Research Announcements of the American Mathematical Society, 2, No. 1, (1996), No. 1, 1-6. 

[6]  Wayne Haga and Sinai Robins, On Kruskals Principle, Proceeding of the Organic Mathematics Workshop, (http://www.cecm.sfu.ca/organics), with an online interactive game by CECM, (1996). 

[5]  [pdf]  Basil Gordon and Sinai Robins, Lacunarity of Dedekind eta-products, Glasgow Math. Journal, 37, (1995), No. 1, 1-14. 

[4]  [pdf]  Ricardo Diaz and Sinai Robins Picks Formula via the Weierstrass P-function, The American Mathematical Monthly, 101, No. 3, (1995), No. 5, 431-437.  

[3]  [pdf]  Ken Ono and Sinai Robins, Superlacunary cusp forms, Proc. Amer. Math. Society, 123, (1995), No. 4, 1021-1029. 

[2]  [pdf]  Ken Ono, Sinai Robins, and Patrick Wahl, On the representations of integers as sums of triangular numbers, Aequationes Math, 50, (1995), No. 1-2, 73-94.  

[1]  [pdf link]  Sinai Robins, Generalized Dedekind eta products, The Rademacher Legacy to Mathematics (University Park, PA 1992) 119-129, Contemporary Math. 166, American Math. Society, Providence, R.I. (1994).  

[0]  [ProQuest LLC]  Sinai Robins, Arithmetic properties of modular forms, Ph.D. Thesis, University of California, Los Angeles, 1991.


Submitted:

[pdf]  Michel Feleiros Martins and Sinai Robins, Sharp inequalities for discrete and continuous muti-tiling, using the    Bombieri-Siegel approach, submitted (2023). 

[pdf] Fabricio Caluza Machado and Sinai Robins, Coefficients of the solid angle and Ehrhart quasi-polynomials, submitted. 

[pdf]  Arnaldo Mandel and Sinai Robins, Dragging the roots of a polynomial to the unit circle, submitted. 

[pdf]  Ricardo Diaz, Quang-Nhat Le and Sinai Robins, Fourier transforms of polytopes, solid angle sums, and discrete volumes, submitted. 

[pdf]  Quang-Nhat Le and Sinai Robins, Macdonald's solid angle sum for real dilations of rational polygons,  submitted. 

[pdf] Christophe Vignat and Sinai Robins, Simple proofs and expressions for the restricted partition function and its polynomial part, submitted.

  

Books:

"Computing the continuous discretely: integer point enumeration in polytopes", Springer 2008, Undergraduate Texts in Mathematics.     [Get the pdf, and see reviews of our book, here]


https://bookstore.ams.org/view?ProductCode=STML/107

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A colloquial description of research papers for the general scientific reader:

Ricardo Diaz, Quang-Nhat Le and Sinai Robins, Fourier transforms of polytopes,  solid angle sums, and discrete volumes,  submitted.

A discretized version of the volume of a polytope may be given by the number of integer points in P.  There are infinite families of discretized volumes of polytopes, but a particularly elegant one is the “solid angle sum” of a polytope, defined as follows.  Place a very small sphere, centered at each lattice point of a given lattice L ⊂ Rd, and consider the proportion of that sphere that intersects the polytope P, called a local solid angle.  Now translate this small sphere to an arbitrary lattice point of L, and again consider the local solid angle contribution, relative to P.  If we sum all of these contributions, over all lattice points of L, we get the "solid angle sum" of P, a nice discrete measure of the volume of P.

It turns out that if we dilate P by an integer (called the dilated polytope tP) and assume that the vertices of P lie on the lattice L, then this solid angle sum is a polynomial in the positive integer parameter t, and this polynomial is traditionally called AP(t) and first studied in detail by I. G. Macdonald.  The coefficients of this polynomial encode certain geometric and number-theoretic properties of the polytope P, but they are still mysterious and not easy to compute in general.

Here we extend the theory of these solid angle sums to all real dilations, for any real polytope (so that its vertices are no longer necessarily on the lattice L).  One of our ingredients is a detailed description of the Fourier transform of the polytope P.  This transform method uses the formula of Stokes, which is a way of integrating by parts in several variables.  Another key tool for us is the Poisson summation formula, applied to smoothings of the indicator function of the polytope P.  It turns out that the combinatorics of the face poset of P plays a central role in the description of the Fourier transform of P, and in keeping track of the infinite series that pop out of Poisson summation.  We also obtain a closed form for the codimension-1 coefficient of the solid angle polynomial AP (t), extending previously known results about this codimension-1 coefficient. 


Undergraduate Publications and informal articles

[1]  Robins, S. The Rotating Coin Puzzle, the UCLA undergraduate Science Journal, 1986.

[2]  Robins, S. The Zen of Mathematics, the Mirror newspaper, UNC, December of 1991, University of Northern Colorado.