[1]  A. Mudgal, Sums of linear transformations in higher dimensions, Q. J. Math. 70 (2019), no. 3, 965–984.  journal 

[2]  A. Mudgal,  Arithmetic combinatorics on Vinogradov systems,  Trans. Amer. Math. Soc. 373 (2020), no. 8, 5491–5516.  journal 

[3]  A. Mudgal, Sum-product estimates for diagonal matrices, Bull. Aust. Math. Soc. 103 (2021), no. 1, 28-37.  journal

[4]  A. Mudgal, Difference sets in higher dimensions,  Math. Proc. Cambridge Philos. Soc. 171 (2021), no. 3, 467-480.  journal

[5]   A. Mudgal, Additive energies on spheres,  J. Lond. Math. Soc. (2) 106 (2022), no. 4, 2927-2958. journal 

[6]  A. Mudgal, New lower bounds for cardinalities of higher dimensional difference sets and sumsets, Discrete Analysis 2022, Paper No. 15, 19 pp. journal 

[7]  A. Mudgal, Diameter free estimates for the quadratic Vinogradov mean value theorem,  Proc. Lond. Math. Soc. (3) 126 (2023), no. 1, 76–128. journal

[8]  B. Cook, K. Hughes, Z. K. Li, A. Mudgal, O. Robert, P. Yung, A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem, Mathematika, 70 (2024), no.1, Paper No. e12231, 32 pp. journal

[9] A. Mudgal, Unbounded expansion of polynomials and products, Mathematische Annalen, accepted, 28 pages. journal

[10]  A. Mudgal, An Elekes-Rónyai theorem for sets with few products, International Mathematics Research Notices, accepted, 17 pages.

[11]  A. Mudgal, Energy estimates in sum-product and convexity problems, submitted, 25 pages. 

[12] Y. Jing,  A. Mudgal, Finding large additive and multiplicative Sidon sets in sets of integers, submitted, 22 pages. 

[13]  Y. Jing, A. Mudgal, Kemperman's inequality and Freiman's lemma via few translates, submitted, 18 pages.

[14]  S. Mansfield, A. Mudgal, A quadratic Vinogradov mean value theorem in finite fields, submitted, 23 pages.

I am mainly interested in topics in Arithmetic Combinatorics and problems lying on the interface of Analytic Number Theory and Harmonic Analysis.  For instance, some of my recent work in [2, 7] considers an arithmetic combinatorial approach to studying the Vinogradov system of diophantine equations, a topic that has seen decisive progress in the past decade via the decoupling programme pioneered by Bourgain-Demeter-Guth, and the efficient congruencing method developed by Wooley. This involves a blend of ideas from incidence geometry and additive combinatorics, and has further applications to discrete restriction estimates for lattice points on spheres [5] as well as a finite field analogue of the quadratic Vinogradov mean value theorem [14].

I am also quite interested in the sum-product problem. In [10], I develop an efficient variant of the many-fold Balog-Szemerédi-Gowers theorem and utilises it to prove a new class of many-fold low energy decompositions, generalising previous work of Bourgain-Chang on many fold sumsets and product sets and answering a question of Balog-Wooley in the integer setting. In joint work with Jing [11], we use these ideas along with various other additive combinatorial and combinatorial geometric techniques to make progress towards a question of Klurman-Pohoata on additive and multiplicative Sidon sets. I have also been able to generalise this circle of ideas to a variety of systems of polynomial equations closely related to Elekes-Rónyai type theorems, see [9,13].

Finally,  I am quite interested in the analysis of sumsets in higher dimensions, a topic that is quite closely connected to the sum-product conjecture as well as Freiman type inverse theorems. Here, I have proved new lower bounds for sums of dilates [1] and difference sets [4,6], answering questions of Balog-Shakan and Ruzsa respectively. In joint work with Jing [12], we further prove a conjecture of Bollobás-Leader-Tiba concerning Cauchy-Davenport type inequalities in the d-dimensional torus. 

My PhD thesis, titled "Arithmetic combinatorics on Vinogradov systems and related topics", is present here.