In this paper, we revisit the square function estimates for extension operators over curves with torsion and prove results for planar curves of the form pT, φpT qq where φpT q is a polynomial of degree at least 2. This includes new results for degenerate, finite type curves endowed with the Euclidean arclength measure such as those given by monomials φpT q " T k for k ě 3. Our results are uniform over all local fields of characteristic 0.
In this paper, we bound the number of solutions to a general Vinogradov system of equations, as well as other related systems, in which the variables are required to satisfy digital restrictions in a given base. Specifically, our sets of permitted digits have the property that there are few representations of a natural number as sums of elements of the digit set-the set of squares serving as a key example. We obtain better bounds using this additive structure than could be deduced purely from the size of the set of variables. In particular, when the digits are required to be squares, we obtain diagonal behaviour with 2k(k + 1) variables.
We show that for integers k ≥ 4 and s ≥ k 2 + (3k − 1)/4, we have an asymptotic formula for the number of solutions, in positive integers, and τ > 0 is sufficiently large. We use Freeman's variant of the Davenport-Heilbronn method, along with a new estimate on the Hardy-Littlewood minor arcs, to obtain this improvement on the original result of Chow.2010 Mathematics Subject Classification. 11D75, 11P05.
In this paper, we bound the number of solutions to a general Vinogradov system of equations [Formula: see text] as well as other related systems, in which the variables are required to satisfy digital restrictions in a given base. Specifically, our sets of permitted digits have the property that there are few representations of a natural number as sums of elements of the digit set — the set of squares serving as a key example. We obtain better bounds using this additive structure than could be deduced purely from the size of the set of variables. In particular, when the digits are required to be squares, we obtain diagonal behavior with [Formula: see text] variables.
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