Number Theory and Fourier Analysis Applications in Physics, Acoustics and Computer Science

Abstract

Number theory is traditionally considered a rather abstract field, far removed from practical applications. In the recent past, however, the “higher arithmetic” has provided highly useful answers to numerous real—world problems. Many of these uses depend on the special correlation and Fourier Transform properties of certain real and complex sequences derived from different branches of number theory, particularly finite fields and quadratic residues. The applications include the design of new musical scales, powerful cryptographic systems, and diffraction gratings for acoustic and electromagnetic waves with unusually broad scatter, with applications in radar camouflage, laser speckle removal, noise abatement, and concert hall acoustics. Another prime domain of number theory is the construction of very effective error—correction codes, such as those used for picture transmission from space vehicles and in compact discs (CDs). Other new uses include schemes for spread—spectrum communication, “error—free” computing, fast computational algorithms, and precision measurements (of interplanetary distances, for example) at extremely low signal—to—noise ratios. In this manner the “fourth prediction” of General Relativity (the slowing of electromagnetic radiation in gravitation fields, predicted by Einstein as early as 1907) has been fully confirmed. In contemporary physics the quasiperiodic route to chaos of nonlinear dynamical systems (the double—pendulum and the three—body problem, to mention two simple examples) are being analyzed in terms of such number theoretic concepts as continued fractions, Fibonacci numbers, the golden mean and Farey trees. Even the recently discovered new state of matter, christened quasicrystals, is most effectively described in terms of arithmetic principles. And last not least, prime numbers, whose distribution combines predictable regularity and surprising randomness, are a rich source of pleasing artistic design - either directly or through the Fourier Transform.

Based on the authors book Number Theory in Science and Communication (Springer-Verlag, 1986)