Papers

My arXiv papers: https://arxiv.org/a/hughes_k_1.html

My Google Scholar profile: https://scholar.google.com/citations?hl=en&user=bdso5a8AAAAJ 

My ORCID profile: 0000-0002-8621-8259

My MR author ID: 962878

Scopus author ID: 54920111500

At this point [19 Mar 2024] preprints of all of my papers appear on the arXiv (link).

Published

1. J. Bober, E. Carneiro, K. Hughes and L. Pierce. On a discrete version of Tanaka's theorem for maximal functions. Proc. Amer. Math. Soc 140 (2012), no. 5, 1669-1680. https://doi.org/10.1090/S0002-9939-2011-11008-6

   • J. Bober, E. Carneiro, K. Hughes, D. Kosz and L. Pierce. Corrigendum to "On a discrete version of Tanaka's theorem for maximal functions". Proc. Amer. Math. Soc. 143 (2015), no. 12, 5471–5473. https://doi.org/10.1090/proc/12778 

     • This corrigendum affected our result about the non-centered Hardy--Littlewood maximal function; we stated our result for one operator but proved it for a similar, yet distinct operator. Thanks to Dariusz Kosz for pointing this out to us and correcting our mistake. Our main result about the centered Hardy--Littlewood maximal function was unaffected.

2. E. Carneiro and K. Hughes.  On the endpoint regularity of discrete maximal operators.  Math. Res. Lett. 19 (2012), no. 6, 1245-1262. https://dx.doi.org/10.4310/MRL.2012.v19.n6.a6

3. K. Hughes, Ben Krause and Bartosz Trojan. The maximal function and square function control the variation: An elementary proof, Proc. Amer. Math. Soc. 144 (2016), 3583-3588. https://doi.org/10.1090/proc/12866

4. K. Hughes, Restricted weak-type endpoint estimates for k-spherical maximal functions, Math. Z. 286 (2017), no. 3-4, 1303-1321. https://doi.org/10.1007/s00209-016-1802-y

   • This paper is a follow-up to my thesis which is Problems and results related to Waring's problem: Maximal functions and ergodic averages.

   • See also this open-access link with enhanced PDF features.

5. K. Hughes, Maximal functions and ergodic averages related to Waring's problem, Isr. J. Math. 217 (2017) no. 1, 17-55. https://doi.org/10.1007/s11856-017-1437-7

   • This paper precedes Restricted weak-type endpoint estimates for k-spherical maximal functions.

   • See also this open-access link with enhanced PDF features.

6. K. Henriot and K. Hughes, Restriction estimates of ε-removal type for k-th powers and paraboloids, Math. Ann. 372, 963–998 (2018). https://doi.org/10.1007/s00208-018-1650-7

7. T. Anderson, B. Cook, K. Hughes and A. Kumchev, Improved ℓ^p-boundedness for integral k-spherical maximal functions, Disc. Anal. (2018). https://doi.org/10.19086/da.3675

8. K. Henriot and K. Hughes, On restriction estimates for discrete quadratic surfaces. Int. Math. Res. Not., Volume 2019, Issue 23, December 2019, 7139-7159. https://doi.org/10.1093/imrn/rnx255

9. K. Hughes, The discrete spherical maximal function over a family of sparse sequences, J. Anal. Math. (2019) 138: 1. https://doi.org/10.1007/s11854-019-0020-z

   • This was the first paper to consider discrete spherical averages along sparse sequences e.g., those with lacunary radii. The goal was to obtain a result for lacunary sequences which went beyond the seminal results of Magyar and Magyar--Stein--Wainger. Instead, I achieved results for structured lacunary sequences of which the only examples I could find were hyper-lacunary. It is an interesting question to find and classify such structured sequences which are lacunary and not faster growing.

   • Since this paper there have been works by Cook and by Kesler--Lacey--Mena which have obtained further results. KLM gave an improved bound for general lacunary sequences following the strategies introduced in this paper and my restricted weak-type paper from 2017 (above). Brian Cook and I gave the same boundedness by a different proof in a more general context. See Bounds for Lacunary maximal functions given by Birch--Magyar averages below.

10. K. Hughes, ℓ^p-improving for discrete spherical averages, Ann. H. Lebesgue, Vol. 3 (2020), 959-980. https://doi.org/10.5802/ahl.50

    • This work was the first to also treat the associated dyadic maximal functions. 

    • See also Kesler--Lacey's subsequent work which obtains sharper bounds for a single average and Kesler's subsequent work on l^p improving for dyadic discrete spherical maximal functions. The 4 dimensional case was first handled in my paper and Kesler obtains results in higher dimensions. Both papers (Kesler and Kesler--Lacey) go on to prove related sparse bounds. 

    • See also Anderson's paper which obtained results for spherical averages whose coordinates are restricted to be primes.

11. T. Anderson, K. Hughes, J. Roos and A. Seeger, Lp to Lq  bounds for spherical maximal operators, Math. Z. 297, 1057-1074 (2021). https://doi.org/10.1007/s00209-020-02546-0

12. B. Cook and K. Hughes, Bounds for lacunary maximal functions given by Birch--Magyar averages. Trans. Amer. Math. Soc. 374 (2021), pp. 3859-3879. https://doi.org/10.1090/tran/8152

    • See also independent work of Kesler--Lacey--Mena who obtained the same range of spaces in the case of spherical averages. Note that KLM use a form of the restricted weak type method which differs from Bourgain/Ionescu by exploiting the lacunary assumption via the union bound; this argument first appeared in my restricted weak type paper above. In fact, I had circulated the proof in the continuous case for several years prior to KLM and discussed this proof and my method in depth with one of the authors at MSRI in Spring 2017.

13. T. Anderson, B. Cook, K. Hughes and A. Kumchev, The ergodic Waring--Goldbach problem, J. Funct. Anal. 282, Issue 5, 1 March 2022, 109334, https://doi.org/10.1016/j.jfa.2021.109334

    • This paper considers a singular variants of Magyar's discrete spherical maximal function by restricting the averages to the prime points on spheres. We obtained sharp results in 7 or more dimensions. The problem remains open in dimensions 5 and 6.

14. S. Dendrinos, K. Hughes and M. Vitturi, Some subcritical estimates for the ℓ^p-improving problem for discrete curves, J. Fourier Anal. Appl. 28, no. 69, 2022. https://doi.org/10.1007/s00041-022-09958-y

    • We employ a discrete version of the method of refinements to study the ℓ^p-improving problem for discrete curves in subcritical regimes of p. Coupling our version of the method of refinements with paucity subconvexity techniques in number theory, we obtain several new bounds. This includes bounds for a single non-linear polynomial, for planar curves of the form (X,f(X)) for f(X) a polynomial of degree at least 2, and for the twisted cubic (X,X^2,X^3). 

15. K. Hughes and T. Wooley, Discrete Restriction for (x,x3) and Related Topics, Int. Math. Res. Not., Volume 2022, Issue 20, October 2022, 15612-15631. https://doi.org/10.1093/imrn/rnab113

    • In this short note we prove an ℓ2 to L10 estimate for the extension (aka adjoint restriction) operator associated to the discrete curve (X,X3). This is interesting, in part, because Demeter has shown that the corresponding putative decoupling inequality fails.

16. J. Brandes and K. Hughes, On the inhomogeneous Vinogradov system, Bull. Aust. Math. Soc., Volume 106, Issue 3, December 2022, 396-403. https://www.doi.org/10.1017/S0004972722000284

    • For most inhomogeneous Vinogradov systems, at even subcritical moments, we obtain an improvement over the nigh-sharp bounds for homogeneous Vinogradov systems due to Bourgain--Demeter--Guth and Wooley. Wooley considered this problem independently and made several improvements; see arXiv:2201.02699, arXiv:2202.05804 and arXiv:2202.14003.

17. B. Cook, K. Hughes and E. Palsson, Supercritical discrete restriction estimates for forms in many variables, Proc. Edinb. Math. Soc., September 2023, 1-17. doi:10.1017/S0013091523000366

    • We prove discrete restriction estimates for a broad class of hypersurfaces and varieties of intermediate codimension. These varieties are far from the purview of decoupling theory and efficient congruencing. As a result, we fall back on Bourgain's circle method approach. 

    • This paper is open access. 

18. B. Cook, K. Hughes, Z. K. Li, A. Mudgal, O. Robert, P.-L. Yung, A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem, Mathematika, Volume 70: e12231, 2024,  https://doi.org/10.1112/mtk.12231

    • Abstract: We interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what precisely does solution counting in older partial progress on Vinogradov's mean value theorem correspond to in Fourier decoupling theory.

    • This paper is open access. 

Accepted and to appear

1. K. Hughes, The pointillist principle for variation operators and jump functions, arXiv:1610.00322, 9pp. https://doi.org/10.33044/revuma.4124

   • This short note extends the results of Moon and Carillo--de Guzman from maximal functions to variations and jump functions.

2. T. C. Anderson, A. Gafni, K. Hughes, R. J. Lemke Oliver, D. Lowry-Duda, F. Thorne, J. Wang, R. Zhang, Improved bounds on number fields of small degree, arXiv:2204.01651, 17pp.

   • In this paper we revisit Schmidt's bound counting the number of number fields of degree n over Q and discriminant bounded by X. We save a small power over Schmidt's bound. Our method injects a few Fourier analytic techniques to the problem which improve bounds in various facets of the problem. The main bottleneck in our approach (and others) lies in an associated sieve problem. 

   • In concurrent work, Bhargava--Shankar--Wang attack the aforementioned, associated sieve problem and obtain a better bound for it. This gives a better improvement than ours. See their work: arXiv:2204.01331.

arXiv Preprints

1. K. Biggs, J. Brandes and K. Hughes, Reinforcing a philosophy: A counting approach to square functions over local fields, arXiv:2201.09649, 18pp.

   • This paper is motivated by the wonderful Reversing a philosophy paper by Gressman--Guo--Pierce--Roos--Yung; arXiv version, published version. In GGPRY, they proved a new, localized square function estimate based on the strong diagonal behavior of a Diophantine inequality. Their method boils down to the mean value theorem which begs the question: do their results hold over other local fields? In non-Archimedean local fields the mean value theorem fails terribly, so this is not obvious. In two dimensions, we show that the answer is yes for polynomial curves. We go futher to prove results for finite type, planar curves, thereby removing GGPRY's non-degenerate hypothesis. Our main method is Wooley's second order differencing method which was used in the above works with Wooley and with Dendrinos+Vitturi. 

   • I submitted this paper to JGEA and it languished there for 17 months without an either an editor or an editor submitting it to be refereed. This was after a technical error in the electronic submission that initially postponed submission by 2 months. Responses from the journal during that time were automatic ones that never addressed the problems. After 19 months altogether, we withdrew this to submit elsewhere.  I have heard of other problems with this journal. 

     • This reflects my experience in handling the submission and does not reflect that of my co-authors. 

2. K. Hughes, Subcritical paucity and ℓ^p-improving estimates for finite-type polynomial curves, arXiv:2201.11468, 12pp. 

3. K. Hughes, Reinforcing a Philosophy: Littlewood-Paley theory for the moment curve, arXiv:2208.07920, 8pp.  

   • This paper relates to the one with a similar title above. In this paper, I give: A simple proof of the boundedness of the square function over the moment curve in generic local fields, and explicit bounds for square functions over real and complex non-degenerate, polynomial curves based on the degree(s) of the curve. The latter argument adapts to fewnomials and Pfaff curves. 

4. K. Hughes, A. Israel and A. Mayeli, On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions, II, arXiv:2403.13092, 18pp.

   • This paper builds on the previous work of Israel [arXiv:1502.04404], Israel--Mayeli [arXiv:2301.09616] and Marceca--Romero--Speckbacher [arXiv:2301.11685]. Using methods from those works we give sharper error bounds than preceding works. 

Other notes

• VKEMS study group, Guiding Principles for Unlocking the Workforce- What Can Mathematics Tell Us?, Working Paper, 2020. ICMS preprint.

• K. Hughes and T. Wooley, Notes on the efficient congruencing method

  • See the attachments at the bottom of the efficient congruencing and decoupling workshop page.