On Waring's Problem With Nonintegral Exponent

Abstract

Thin films of collodion and Canada balsam have been prepared on water surfaces. Their thickness has been evaluated interferometrically over the range 4-s~ h and the relation between concentration and thickness studied. The effect of temperature on film thickness has been evaluated at 25, 100 and 150" c.

Additive bases arising from functions in a Hardy field

“…of integer parts of c th powers is an asymptotic basis. This question has been studied further by Deshouillers [4] and by Arkhipov and Zhitkov [1], among others. Whereas those authors focused on the order of the basis (i.e., the number of unknowns in an equation analogous to (1.1) that ensure solvability), in this work we ask the general question if the sequence [n c ] can be replaced by other sequences.…”

Exceptional set of a representation with fractional powers

“…It follows from the work of B. I. Segal [17] that G(c) is finite, and in fact, Segal obtained an upper bound for G(c). For further developments and improvements, see [1,2,7].…”

Waring's Problem With Piatetski‐shapiro Numbers

“…for a fixed k 2 and c > 1. In this setting it is natural to ask for the smallest number of summands s = G k (c) that would be needed for a given c. Indeed, for k = 1, this has been done by several authors (see [1,3,8,10,11]). Unfortunately, for k 2, we cannot adapt the methods used therein for an arbitrary c > 1.…”

“…In the theorem below, instead of all sufficiently large integers, we require that almost all integers be represented by (1). THEOREM 3.…”

“…(s+2t)δ−k . Making the change of variable yN = x k in I (β) and J (β), and then substituting γ = βN in I(m), we see thatI(m) = (δ/k) s+2t N (s+2t)δ/k−1 lim λ→∞ R φ(u) λ −λ e(γ (u − mN −1 )) dγ du, where φ(u) = · · · y 1 ,...,y 2t ∈[θ,1] y 2t+1 ,...,y 2t+s−1 ∈[0,1] , . .…”