Fernando Xuancheng Shao



E-mail: xuancheng dot shao at uky dot edu

Department website

CV

I am an Associate Professor of Mathematics at University of Kentucky. Before joining UK in 2017, I had s postdoctoral position at University of Oxford, supported by a Glasstone Research Fellowship. I received my PhD in 2014 from Stanford University under the supervision of Kannan Soundararajan.

My research interests are in analytic number theory and arithmetic combinatorics. I'm interested in various questions concerning character and exponential sums, sieve theory, smooth numbers, Szemeredi's theorem, additive problems in primes, structures in sumsets, the theory of Gowers norms and nilsequences, etc.

Papers

  1. The sum-product phenomenon for dense subsets of finite fields.
          Submitted. pdf

  2. (with J. Lim, A. Mudgal, C. Pohoata) On exponential Freiman dimension.
          Submitted. pdf

  3. (with A. Alsetri) Burgess-type character sum estimates over generalized arithmetic progressions of rank 2.
         Bull. Lond. Math. Soc. 58 (2026), no. 2, Paper No. e70293, 16 pp. pdf

  4. (with M. Wang) Quantitative bounds in a popular polynomial Szemeredi theorem.
          Submitted. pdf

  5. (with I. Shparlinksi, L. Wijaya) Sums of Kloosterman sums over square-free and smooth integers.
          Submitted. pdf

  6. (with K. Matomäki, M. Radziwiłł, T. Tao, J. Teräväinen) Higher uniformity of arithmetic functions in short intervals II. Almost all intervals.
          Submitted. pdf

  7. (with S. Peluse, S. Prendiville) Bounds in a popular multidimensional nonlinear Roth theorem.
         J. Lond. Math. Soc. (2) 110 (2024), no. 5, Paper No. e70019, 35 pp. pdf

  8. Additive energies of subsets of discrete cubes.
          Submitted. pdf

  9. (with A. Alsetri) Density versions of the binary Goldbach problem.
         Acta Arith. 218 (2025), no. 3, 285-295. pdf

  10. (with K. Matomäki, T. Tao, J. Teräväinen) Higher uniformity of arithmetic functions in short intervals I. All intervals.
         Forum Math. Pi 11 (2023), Paper No. e29, 97 pp. pdf

  11. (with A. Alsetri) On Hilbert cubes and primitive roots in finite fields.
         Arch. Math. (Basel) 118 (2022), no. 1, 49-56. pdf

  12. (with P. V. Bienvenu, J. Teräväinen) A transference principle for systems of linear equations, and applications to almost twin primes.
         Algebra Number Theory 17 (2023), no. 2, 497-539. pdf

  13. (with K. Matomäki, M. Radziwiłł, T. Tao, J. Teräväinen) Singmaster's conjecture in the interior of Pascal's triangle.
         Q. J. Math. 73 (2022), no. 3, 1137-1177. pdf

  14. (with J. Teräväinen) The Bombieri-Vinogradov theorem for nilsequences.
         Discrete Anal. 2021, Paper No. 21, 55 pp. pdf

  15. (with K. Matomäki) Discorrelation between primes in short intervals and polynomial phases.
         Int. Math. Res. Not. IMRN 2021, no. 16, 12330-12355. pdf

  16. On an almost all version of the Balog-Szemeredi-Gowers theorem.
         Discrete Anal. 2019, Paper No. 12, 18 pp. pdf

  17. (with W. Xu) A robust version of Freiman's 3k-4 Theorem and applications.
         Math. Proc. Cambridge Philos. Soc. 166 (2019), no. 3, 567-581. pdf

  18. (with A. Granville) When does the Bombieri-Vinogradov Theorem hold for a given multiplicative function?
         Forum Math. Sigma 6 (2018), e15, 23 pp. pdf

  19. (with S. Drappeau and A. Granville) Smooth-supported multiplicative functions in arithmetic progressions beyond the x^{1/2}-barrier.
         Mathematika 63 (2017), no. 3, 895--918. pdf

  20. (with A. Granville) Bombieri-Vinogradov for multiplicative functions, and beyond the x^{1/2}-barrier.
         Adv. Math. 350 (2019), 304-358. pdf

  21. (with K. Matomäki and J. Maynard) Vinogradov's theorem with almost equal summands.
         Proc. Lond. Math. Soc. (3) 115 (2017), no. 2, 323-347. pdf

  22. (with B. B. Bhattacharya, S. Ganguly, and Y. Zhao) Upper tail large deviations for arithmetic progressions in a random set.
         Int. Math. Res. Not. IMRN 2020, no. 1, 167-213. pdf

  23. (with K. Matomäki) When the sieve works II.
         J. Reine Angew. Math. 763 (2020), 1-24. pdf

  24. Gowers norms of multiplicative functions in progressions on average.
         Algebra Number Theory 11, 961-982 (2017). pdf

  25. Narrow arithmetic progressions in the primes.
         Int. Math. Res. Not. IMRN, 391-428 (2017). pdf

  26. (with K. Matomäki) Vinogradov's three primes theorem with almost twin primes.
         Compos. Math. 153, 1220-1256 (2017). pdf

  27. (with S. Drappeau) Weyl sums, mean value estimates, and Waring's problem with friable numbers.
         Acta Arith. 176, 249-299 (2016). pdf

  28. On an inverse ternary Goldbach problem.
         Amer. J. Math. 138, 1167-1191 (2016). pdf

  29. Polynomial values modulo primes on average and sharpness of the larger sieve.
         Algebra Number Theory 9, 2325-2346 (2015). pdf

  30. Character sums over unions of intervals.
         Forum Math. 27, 3017-3026 (2015). pdf

  31. Finding linear patterns of complexity one.
         Int. Math. Res. Not. IMRN, 2311-2327 (2015). pdf

  32. An L-function-free proof of Vinogradov's three primes theorem.
         Forum Math. Sigma 2, e27, 26 pp (2014). pdf

  33. (with P. Diaconis and K. Soundararajan) Carries, group theory, and additive combinatorics.
         Amer. Math. Monthly 121, 674-688 (2014). pdf

  34. Large values of the additive energy in R^d and Z^d.
         Math. Proc. Cambridge Philos. Soc. 156, 327-341 (2014). pdf

  35. A density version of the Vinogradov three primes theorem.
         Duke Math. J. 163, 489-512 (2014). pdf

  36. On character sums and exponential sums over generalized arithmetic progressions.
         Bull. Lond. Math. Soc. 45, 541-550 (2013). pdf

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