Research

Research articles published:

(Live links to the pdf files are highlighted in green letters below.  More expository descriptions of some of my articles are available at the bottom of this page)

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

[47]  Fabricio Caluza Machado and Sinai Robins,  The null set of a polytope and the Pompeiu property for polytopes, Journal d'Analyse 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]  Sinai Robins, Ehrhart quasi-polynomials via Barnes polynomials  and discrete moments of parallelepipeds, submitted (2026). 

[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:

1.  (Joint with Matthias Beck), published in the Springer Undergraduates Texts series:

"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]

2.  Fourier analysis on polytopes, and the geometry of numbers, part I: a friendly introduction, published in 2024, by the American Mathematical Society book series "the student mathematical library":

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

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

Sinai Robins, Ehrhart quasi-polynomials via Barnes polynomials  and discrete moments of parallelepipeds, submitted (2026). 

A discretized volume of a polytope P whose vertices have integer coordinates (integer polytope)  is defined by the number of integer points in P.   In the 1950's Eugene Ehrhart discovered that the discrete volume of the k'th integer dilate of P (called kP) is a polynomial function of the dilation parameter k, called the Ehrhart polynomial of P.  Ehrhart also discovered a similar phenomenon for polytopes whose vertcies have rational coordinates (rational polytopes):  the number of integer points in kP is precisely a quasi-polynomial in k. 

Since Ehrhart's discoveries, most of the coefficients of these Ehrhart polynomials and Ehrhart quasi-polynomials have remained mysterious.. They capture crucial geometric and topological information about P.   For example, if one extends Ehrhart to study non-convex polytopes P, then the constant term of the ensuing Ehrhart polynomial is the Euler characteristic of P.  

Here we give new and very explicit formulas for the Ehrhart quasi-polynomial of any simple rational polytope, in terms of two new ingredients: Barnes polynomials and discrete moments of half-open parallelepipeds.  A d-dimensional polytope is called simple if each of its vertices has exactly d edges incident with it.   These formulas hold to all rational polytopes, and any positive dilate of it.   

One new aspect here is the appearance of an intriguing and general z-parameter, which is guaranteed to disappear once we use it to compute the coefficients of any quasi-polynomial. 

There is a large body of literature concerned with moments of a rational polytope P.  Here we give new formulas for such moments, in any positive dilate of P,  again in terms of Barnes polynomials and discrete moments of integer parallelepipeds.   The appearance of the Barnes polynomials and of the Barnes numbers (which are the constant terms of the Barnes polynomials)  allow for very explicit computations.   These polynomials have been studied mostly in contexts distinct from this one, such as Barnes zeta functions, and number-theoretic contexts, such as Dedekind sums, that exploit their inherent connection with the classical Bernoulli polynomials.  

This paper shows that the complexity of computing Ehrhart quasi-polynomials lies mainly in the computation of various discrete moments of parallelepipeds.  These discrete moments are finite sums over a particular translate of a lattice (a lattice flow) that is contained in an integer parallelepiped.   

Some of the consequences of these results involve new, canonical vanishing identities for rational polytopes, generalizing the work of Michel Brion from 1988.  Another consequence is a differential equation for the coefficients of discrete volumes, and of discrete moments of a rational polytope, generalizing the work of Eva Linke from 2011. 

An integer polytope P is called smooth if for each of its vertices v, the edges incident with v form a basis for the integer lattice Z^d.   For smooth polytopes, there is a dramatic simplification of our formulas, and we therefore obtain new and much simpler formulations of Ehrhart quasi-polynomials, as well as simplified formulas for discrete moments and for vanishing identities.   

From the perspective of Barnes polynomials, these results give a new playground for their interactions, and their reciprocity laws.   In other words, these formulations show the utility of Barnes polynomials in geometric combinatorics, due to their very rich structure that extend the classical 1-dimensional Bernoulli polynomials.  

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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 defined 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.