SMALL FRACTIONAL-PARTS OF ADDITIVE FORMS

We show how the methods of Vaughan & Wooley, which have proved fruitfu l in dealing with Waring's problem, may also be used to investigate th e fractional parts of an additive form. Results are obtained which are near to best possible for forms with algebraic coefficients. New resu lts are also obtained in the general case. Extensions are given to mak e several additive forms simultaneously small. The key ingredients in this work are: mean value theorems for exponential sums, the use of a small common factor for the integer variables, and the large sieve ine quality.