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Short interval results for powerfree polynomials over finite fields

https://doi.org/10.1142/S1793042124500453

Kumchev, Angel V; McNew, Nathan G; Park, Ariana (April 2024, International Journal of Number Theory)

Let [Formula: see text] be an integer and [Formula: see text] be a finite field with [Formula: see text] elements. We prove several results on the distribution in short intervals of polynomials in [Formula: see text] that are not divisible by the [Formula: see text]th power of any non-constant polynomial. Our main result generalizes a recent theorem by Carmon and Entin on the distribution of squarefree polynomials to all [Formula: see text]. We also develop polynomial versions of the classical techniques used to study gaps between [Formula: see text]-free integers in [Formula: see text]. We apply these techniques to obtain analogs in [Formula: see text] of some classical theorems on the distribution of [Formula: see text]-free integers. The latter results complement the main theorem in the case when the degrees of the polynomials are of moderate size.
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A framework for discrete bilinear spherical averages and applications to $$\ell ^p$$-improving estimates

https://doi.org/10.4064/cm9216-1-2024

Anderson, Theresa C.; Kumchev, Angel V.; Palsson, Eyvindur A. (January 2024, Colloquium Mathematicum)

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Discrete Maximal Operators Over Surfaces of Higher Codimension

https://doi.org/10.1007/s44007-021-00017-4

Anderson, Theresa C.; Kumchev, Angel V.; Palsson, Eyvindur A. (June 2022, La Matematica)

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Hilbert transforms and the equidistribution of zeros of polynomials

https://doi.org/10.1016/j.jfa.2021.109199

Carneiro, Emanuel; Das, Mithun Kumar; Florea, Alexandra; Kumchev, Angel V.; Malik, Amita; Milinovich, Micah B.; Turnage-Butterbaugh, Caroline; Wang, Jiuya (November 2021, Journal of Functional Analysis)
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