Three Additive Problems

Abstract

Tiny dew droplets deposited on a copper plate were controlled constantly by a developed control technique using scattered laser light for studying initial dropwise condensation. The technique employs proportional control combined with shifting movement by an integrator. The droplets were controlled for 60 min at almost constant diameters in a range from only a few micrometers to tens of micrometers and were almost hemispherical in the initial condensation at room temperature.

Exceptional set of a representation with fractional powers

“…It can be conjectured that in this case G(c) = 2. This was proved by J. M. Deshouillers [2] for the range 1 < c < 4/3, and by S. A. Gritsenko [7] for 1 < c < 55/41. The proofs are based on the method of trigonometric sums.…”

“…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].…”

Weak type (1,1) estimates for a class of discrete rough maximal functions

“…This is closely connected to the problem of representation of a given integer as a difference of two numbers of the form [m α ]. In order to obtain necessary estimates, we use B. I. Segals approach, [7], [9], see also [6]. The p (1 < p ≤ ∞) boundedness of the maximal function M * has been established in [1,3] and [2].…”

A Binary Additive Equation Involving Fractional Powers

“…with integers m 1 , m 2 ; henceforth, [θ] denotes the integral part of θ. Subsequently, the range for c in this result was extended by Gritsenko [3] and Konyagin [5]. In particular, the latter author showed that (1) has solutions in integers m 1 , m 2 for 1 < c < 3 2 and n sufficiently large.…”