The theory of multiple trigonometric sums (also called “exponential sums”), as constructed systematically in this monograph, has reached the same degree of completion as the theory of one-dimensional trigonometric sums. Compared to the one-dimensional case, numerous new effects appear because there is a wide variety both of domains of the summation variables and of functions in the exponent. The first nine chapters of the monograph are the translation of the original Russian book (1987, Nauka, Moscow), while chapters 10 to 12 are devoted to several new applications of trigonometric sums and integrals to problems of number theory and analysis: estimates of trigonometric (oscillating) integrals and applications of the p-adic method in estimating trigonometric sums and in solving additive problems, including the Waring problem and the Artin conjecture on a local representation of zero by a form. The book will be useful for a wide range of mathematicians.

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