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 study arithmetic questions involving restricted-digit and Cantor-type sets. In Primes with Restricted-Digit Differences, joint with Rui Han and Fan Yang, we investigate pairs of primes whose difference lies in a restricted-digit set, and three-term arithmetic progressions in primes whose common difference lies in such a set. Under a digit non-resonance condition, we establish a localized Fourier criterion for asymptotic counting formulas with explicit leading constants. A transfer-operator argument verifies this criterion for several families of digit sets in explicit parameter ranges.

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 parameter-free, scale-invariant ℓ₁/ℓ₂ model, analyzes conditions under which sparse vectors are local minimizers, and develops numerical methods for the constrained problem. Applying this model to image gradients also gives a proof-of-concept example in MRI reconstruction.

Our accelerated schemes reduce computation time and perform particularly well in experiments with signals of high dynamic range. A complementary lifted ℓ₁ framework encompasses several sparsity models and provides equivalence and convergence guarantees. This part of my research reflects my interest in using mathematical ideas to address practical recovery problems.