Research
My work uses analytic and optimization methods to study arithmetic structure and inverse problems, spanning number theory, harmonic analysis, additive combinatorics, and sparse signal recovery.
Analytic number theory and primes
I study questions about prime numbers using the Hardy–Littlewood circle method and related Fourier-analytic tools. A central goal is to understand how the distribution of the primes controls averaging operators and additive representations.
Discrete harmonic analysis
Much of my work concerns discrete averages over sparse arithmetic sets. This includes sharp improving estimates, maximal inequalities, and sparse bounds for averages over the primes, together with transference and interpolation methods adapted to arithmetic operators.
Additive problems and arithmetic progressions
I am interested in Goldbach-type questions, density versions of additive prime problems, and averages over primes in arithmetic progressions. Joint work on Gaussian primes studies both additive representations and improving estimates in sectors, while related work establishes density results for higher-order sums of primes.
Restricted-digit and Cantor-type arithmetic sets
I am also interested in arithmetic questions for sets defined by digit restrictions, including missing-digit and Cantor-type subsets of the integers. These problems naturally involve the Fourier structure of digit measures and major-arc/minor-arc decompositions.
Sparse signal recovery and nonconvex optimization
I also study the recovery of sparse signals from undersampled measurements through nonconvex regularization and optimization. My joint work develops the scale-invariant ℓ1/ℓ2 model and analyzes conditions for sparse recovery, together with efficient and accelerated algorithms for solving the resulting minimization problems. A complementary lifted ℓ1 framework encompasses several sparsity models and provides equivalence and convergence guarantees. See the papers on the scale-invariant model, accelerated schemes, and lifted framework.