@article{21706,
  abstract     = {Consider a quadratic polynomial Q(ξ1, . . . , ξn) of independent Rademacher random variables ξ1, . . . , ξn. To what extent can Q(ξ1, . . . , ξn) concentrate on a single value? This quadratic version of the classical Littlewood–Offord problem was popularised by Costello, Tao and Vu in their study of symmetric random matrices. In this paper, we obtain an essentially optimal bound for this problem, as conjectured by Nguyen and Vu. Specifically, if Q(ξ1, . . . , ξn) ‘robustly depends on at least m of the ξi’ in the sense that there is no way to pin down the value of Q(ξ1, . . . , ξn) by fixing values for fewer than m of the variables ξi, then we have Pr[Q(ξ1, . . . , ξn) = 0] ≤ O(1/√m). This also implies a similar result in the case where ξ1, . . . , ξn have arbitrary distributions. Our proof combines a number of ideas that may be of independent interest, including an inductive decoupling scheme that reduces quadratic anticoncentration problems
to high-dimensional linear anticoncentration problems. Also, one application of our main result is the resolution of a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn related to graph inducibility. },
  author       = {Kwan, Matthew Alan and Sauermann, Lisa},
  issn         = {1570-5846},
  journal      = {Compositio Mathematica},
  number       = {12},
  pages        = {3089--3139},
  publisher    = {Cambridge University Press},
  title        = {{Resolution of the quadratic Littlewood–Offord problem}},
  doi          = {10.1112/S0010437X25102789},
  volume       = {161},
  year         = {2025},
}

@article{10711,
  abstract     = {In this paper, we investigate the distribution of the maximum of partial sums of families of  m -periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of  ℓ -adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of  m -periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp.},
  author       = {Autissier, Pascal and Bonolis, Dante and Lamzouri, Youness},
  issn         = {1570-5846},
  journal      = {Compositio Mathematica},
  keywords     = {Algebra and Number Theory},
  number       = {7},
  pages        = {1610--1651},
  publisher    = {Cambridge University Press},
  title        = {{The distribution of the maximum of partial sums of Kloosterman sums and other trace functions}},
  doi          = {10.1112/s0010437x21007351},
  volume       = {157},
  year         = {2021},
}

@article{264,
  abstract     = {Given a family of varieties over a number field, we determine conditions under which there is a Brauer-Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.},
  author       = {Bright, Maritn and Browning, Timothy D and Loughran, Daniel},
  issn         = {1570-5846},
  journal      = {Compositio Mathematica},
  number       = {7},
  pages        = {1435 -- 1475},
  publisher    = {Cambridge University Press},
  title        = {{Failures of weak approximation in families}},
  doi          = {10.1112/S0010437X16007405},
  volume       = {152},
  year         = {2016},
}