About

I am the Sarofim-Rock Professor of Business Administration in the Entrepreneurial Management Unit at Harvard Business School; a Faculty Affiliate of the Harvard Department of Economics and the Harvard Center of Mathematical Sciences and Applications; and an a16z crypto Research Partner. This semester, I am visiting the Technological Innovation, Entrepreneurship, and Strategic Management (TIES) Group at the MIT Sloan School of Management.

Research topics

• Microeconomics
• Economics
• Business
• Political Science
• Law and economics
• Industrial organization
• Market economy
• Monetary economics
• Virology
• Psychology
• Marketing
• Law
• Mathematical economics
• Biology
• Medicine
• Neoclassical economics
• Finance
• Internal medicine
• Positive economics

Selected publications

• Simultaneous Niven Numbers in Arithmetic Progressions for Power-Related Bases

  Open MIND · 2026-02-01

  preprint1st authorCorresponding

  Recently, Harrington, Litman, and Wong [Bulletin of the Australian Mathematical Society, 2024; arXiv:2303.06534] proved that every arithmetic progression contains infinitely many base-$b$ Niven numbers, for any fixed $b\ge 2$. We use a sparse repunit construction to treat a structured two-base version of the same problem, showing that every arithmetic progression with common difference relatively prime to $b$ contains infinitely many integers that are simultaneously $b$-Niven and $b^k$-Niven (indeed, we can obtain simultaneous $b^\ell$-Niven-ness for $\ell=1,\ldots, k$).

  DOI

• SIMULTANEOUS NIVEN NUMBERS IN ARITHMETIC PROGRESSIONS FOR POWER-RELATED BASES

  Bulletin of the Australian Mathematical Society · 2026-04-21

  articleOpen access1st authorCorresponding

  Abstract Recently, Harrington et al. [‘Every arithmetic progression contains infinitely many b -Niven numbers’, Bull. Aust. Math. Soc. 109 (3) (2024), 409–413] proved that every arithmetic progression contains infinitely many base- b Niven numbers for any fixed $b\ge 2$ . We use a sparse repunit construction to treat a structured two-base version of the same problem, showing that every arithmetic progression with common difference relatively prime to b contains infinitely many integers that are simultaneously b -Niven and $b^k$ -Niven (indeed, we can obtain simultaneous $b^\ell $ -Niven-ness for $\ell =1,\ldots , k$ ).

  PublisherDOI

• Bitcoin Makes Time Travel Possible

  SSRN Electronic Journal · 2026-01-01

  preprintOpen accessSenior author
  PublisherDOI

• Simultaneous Niven Numbers in Arithmetic Progressions for Power-Related Bases

  ArXiv.org · 2026-02-01

  articleOpen access1st authorCorresponding

  Recently, Harrington, Litman, and Wong [Bulletin of the Australian Mathematical Society, 2024; arXiv:2303.06534] proved that every arithmetic progression contains infinitely many base-$b$ Niven numbers, for any fixed $b\ge 2$. We use a sparse repunit construction to treat a structured two-base version of the same problem, showing that every arithmetic progression with common difference relatively prime to $b$ contains infinitely many integers that are simultaneously $b$-Niven and $b^k$-Niven (indeed, we can obtain simultaneous $b^\ell$-Niven-ness for $\ell=1,\ldots, k$).

  PublisherOA PDF

• A Fixed-Prime Criterion for Reciprocals in Missing-Digit Sets

  arXiv (Cornell University) · 2026-04-13

  articleOpen access1st authorCorresponding

  We prove a structural upper bound on the $p$-adic valuation of denominators of rationals belonging to a missing-digit set $K_{m,D}$, generalizing a key step in recent work of Lin, Wu, and Yang [arXiv:2603.24614] on reciprocals of factorials. For a rational $\frac{r}{Q}$ with $\gcd(Q,m)=1$ and a fixed prime $p_0\nmid m$, membership in $K_{m,D}$ forces $ν_{p_0}(Q)$ to be controlled by the $p_0$-adic valuation of the multiplicative order of $m$ modulo the radical of $Q$, with explicit overhead depending only on $m$ and $D$. Because the obstruction is stated at the level of a single denominator, a pair of sequence-specific valuation estimates converts it into an effective finiteness criterion for $\left\{\frac{1}{a_n}:n\in\mathbb{N}\right\}\cap K_{m,D}$. Specializing to the case in which $Q$ is the part of $n!$ coprime to $m$ recovers the fixed-prime step in the Lin--Wu--Yang argument. As applications, we treat reciprocals of superfactorials, products of polynomial values, and products of Fibonacci numbers. We also exhibit an exponential family -- products of $(m^k-1)$ -- for which the full structural criterion applies but a coarser largest-prime-factor formulation does not.

  PublisherOA PDF

• An improved bound for sumsets of thick compact sets via the Shapley--Folkman theorem

  arXiv (Cornell University) · 2026-04-06

  preprintOpen access1st authorCorresponding

  Let $E_1,\dots,E_n \subset \mathbb{R}^d$ be compact sets of positive diameter with Feng--Wu thickness at least $c>0$. Feng and Wu proved that $E_1+\cdots+E_n$ has non-empty interior when $n>2^{11}c^{-3}+1$. We show that \[n>\frac{\sqrt d}{(\sqrt{1+c}-1)^2}=\frac{\sqrt d\,(\sqrt{1+c}+1)^2}{c^2}\] already suffices. In particular, since $06\sqrt d\,c^{-2}$ is enough. For fixed dimension $d$, this improves the exponent in $c^{-1}$ from $3$ to $2$, while introducing only an explicit factor of $\sqrt d$. The proof replaces the one-summand-at-a-time enlargement of Feng--Wu by a simultaneous convexification step based on a radius form of the Shapley--Folkman theorem.

  PublisherDOI

• A Fixed-Prime Criterion for Reciprocals in Missing-Digit Sets

  arXiv (Cornell University) · 2026-04-13

  preprintOpen access1st authorCorresponding

  We prove a structural upper bound on the $p$-adic valuation of denominators of rationals belonging to a missing-digit set $K_{m,D}$, generalizing a key step in recent work of Lin, Wu, and Yang [arXiv:2603.24614] on reciprocals of factorials. For a rational $\frac{r}{Q}$ with $\gcd(Q,m)=1$ and a fixed prime $p_0\nmid m$, membership in $K_{m,D}$ forces $ν_{p_0}(Q)$ to be controlled by the $p_0$-adic valuation of the multiplicative order of $m$ modulo the radical of $Q$, with explicit overhead depending only on $m$ and $D$. Because the obstruction is stated at the level of a single denominator, a pair of sequence-specific valuation estimates converts it into an effective finiteness criterion for $\left\{\frac{1}{a_n}:n\in\mathbb{N}\right\}\cap K_{m,D}$. Specializing to the case in which $Q$ is the part of $n!$ coprime to $m$ recovers the fixed-prime step in the Lin--Wu--Yang argument. As applications, we treat reciprocals of superfactorials, products of polynomial values, and products of Fibonacci numbers. We also exhibit an exponential family -- products of $(m^k-1)$ -- for which the full structural criterion applies but a coarser largest-prime-factor formulation does not.

  PublisherDOI

• Squared edge lengths of regular simplices with rational vertices

  ArXiv.org · 2026-05-12

  articleOpen access1st authorCorresponding

  We determine exactly which positive rational numbers occur as squared edge lengths of regular $d$-simplices with vertices in $\mathbb{Q}^n$. The answer exhibits a sharp stabilization phenomenon: once $n-d\geq 3$, every positive rational number occurs, while codimensions $0$, $1$, and $2$ are governed by explicit square-class, norm-group, and Hilbert-symbol conditions. The proof reduces simplex realizability to the Hasse--Minkowski classification of rational quadratic forms.

  PublisherOA PDF

• How Thick Is the Sierpiński Triangle?

  arXiv (Cornell University) · 2026-05-02

  preprintOpen access1st authorCorresponding

  Although the Sierpiński triangle has planar area $0$, it is uniformly non-flat: at every point and every scale, its nearby points span a two-dimensional region of comparable size. We prove a sharp version of this statement, showing that the Feng--Wu thickness of $E$ is exactly $\sqrt{3}/6$, the inradius of a unit equilateral triangle. More precisely, if $E$ is the standard Sierpiński triangle of side length $1$ and $B(x,r)$ denotes the closed disk of radius $r$ centered at $x$, then for every $x\in E$ and every $0

  PublisherDOI

• An improved bound for sumsets of thick compact sets via the Shapley--Folkman theorem

  arXiv (Cornell University) · 2026-04-06

  articleOpen access1st authorCorresponding

  Let $E_1,\dots,E_n \subset \mathbb{R}^d$ be compact sets of positive diameter with Feng--Wu thickness at least $c>0$. Feng and Wu proved that $E_1+\cdots+E_n$ has non-empty interior when $n>2^{11}c^{-3}+1$. We show that \[n>\frac{\sqrt d}{(\sqrt{1+c}-1)^2}=\frac{\sqrt d\,(\sqrt{1+c}+1)^2}{c^2}\] already suffices. In particular, since $0<c\le 1$, the bound $n>6\sqrt d\,c^{-2}$ is enough. For fixed dimension $d$, this improves the exponent in $c^{-1}$ from $3$ to $2$, while introducing only an explicit factor of $\sqrt d$. The proof replaces the one-summand-at-a-time enlargement of Feng--Wu by a simultaneous convexification step based on a radius form of the Shapley--Folkman theorem.

  PublisherOA PDF

Recent grants

• Preferences in Matching Market Design

  NSF · $221k · 2015–2021

Frequent coauthors

• Lauren Cohen

  104 shared

• Umit G. Gurun

  101 shared

• John William Hatfield

  45 shared

• Edward L. Glaeser

  National Bureau of Economic Research

  24 shared

• Tayfun Sönmez

  Boston College

  22 shared

• Daniel M. Kane

  21 shared

• Parag A. Pathak

  17 shared

• Zachary Abel

  17 shared

Education

• B.A., Mathematics (with a minor in Ethnomusicology)

  Harvard University

  2009

• M.A.

  Harvard University

  2010

• Ph.D., Business Economics

  Harvard University

  2011