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More on inhomogeneous diophantine approximation

Christopher G. Pinner — 2001

Journal de théorie des nombres de Bordeaux

For an irrational real number α and real number γ we consider the inhomogeneous approximation constant M ( α , γ ) : = lim inf | n | | n | | | n α - γ | | via the semi-regular negative continued fraction expansion of α α = 1 a 1 - 1 a 2 - 1 a 3 - and an appropriate alpha-expansion of γ . We give an upper bound on the case of worst inhomogeneous approximation, ρ ( α ) : = sup γ 𝐙 + α 𝐙 M ( α , γ ) , which is sharp when the partial quotients ai are almost all even and at least four. When the negative expansion...

A system of simultaneous congruences arising from trinomial exponential sums

Todd CochraneJeremy CoffeltChristopher Pinner — 2006

Journal de Théorie des Nombres de Bordeaux

For a prime p and positive integers < k < h < p with d = ( h , k , , p - 1 ) , we show that M , the number of simultaneous solutions x , y , z , w in p * to x h + y h = z h + w h , x k + y k = z k + w k , x + y = z + w , satisfies M 3 d 2 ( p - 1 ) 2 + 25 h k ( p - 1 ) . When h k = o ( p d 2 ) we obtain a precise asymptotic count on M . This leads to the new twisted exponential sum bound x = 1 p - 1 χ ( x ) e 2 π i f ( x ) / p 3 1 4 d 1 2 p 7 8 + 5 h k 1 4 p 5 8 , for trinomials f = a x h + b x k + c x , and to results on the average size of such sums.

Waring's number for large subgroups of ℤ*ₚ*

Todd CochraneDerrick HartChristopher PinnerCraig Spencer — 2014

Acta Arithmetica

Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain t = p 22 / 39 + ϵ , t = p 15 / 29 + ϵ and for s ≥ 6, t = p ( 9 s + 45 ) / ( 29 s + 33 ) + ϵ . For s ≥ 24 further improvements are made, such as t 32 = p 5 / 16 + ϵ and t 128 = p 1 / 4 .

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