Harmonic Sieve: Integrating Fourier Analysis with Sieve Theory for Additive Problems

Although Kolountzakis (1995) and Bourgain (1990) touch on the interface between harmonic analysis and additive number theory, a systematic integration with sieve methods is lacking. This idea proposes a “harmonic sieve,” wherein sieve weights and filtering procedures are optimized using Fourier analytic tools. For instance, one could use the spectral properties of additive sets (like having large or small Fourier coefficients) to guide or adjust the sieve, thereby gaining finer control over the distribution of sumsets or solutions to additive equations. This could be particularly powerful for binary additive problems over finite fields (see Sawin, Shusterman, Kowalski, 2022), where Fourier methods are already fruitful. Innovating at this intersection may lead to breakthroughs in bounding or characterizing exceptional sets and could shed new light on phenomena like uniformity and pseudorandomness in number-theoretic sequences.

References:

1. BINARY ADDITIVE PROBLEMS FOR POLYNOMIALS OVER FINITE FIELDS. W. Sawin, M. Shusterman, E. Kowalski (2022).
2. Probabilistic and constructive methods in harmonic analysis and additive number theory. Mihail N. Kolountzakis (1995).
3. Problems of almost everywhere convergence related to harmonic analysis and number theory. J. Bourgain (1990).
4. BINARY ADDITIVE PROBLEMS FOR POLYNOMIALS OVER FINITE FIELDS. W. Sawin, M. Shusterman, E. Kowalski (2022).