Small Fractional Parts of Polynomials

Features notes that starts out with Heilbronn's Theorem on quadratic polynomials. This title deals with arbitrary polynomials with constant term zero. It takes up simultaneous approximation of quadratic polynomials. It discusses special quadratic polynomials in several variables.

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Knowledge about fractional parts of linear polynomials is fairly satisfactory. Knowledge about fractional parts of nonlinear polynomials is not so satisfactory. In these notes, the author starts out with Heilbronn's Theorem on quadratic polynomials and branches out in three directions. In Sections 7-12, he deals with arbitrary polynomials with constant term zero. In Sections 13-19, he takes up simultaneous approximation of quadratic polynomials. In Sections 20-21, he discusses special quadratic polynomials in several variables. There are many open questions: in fact, most of the results obtained in these notes are almost certainly not best possible. Since the theory is not in its final form including the most general situation, i.e. simultaneous fractional parts of polynomials in several variables of arbitrary degree. On the other hand, he has given all proofs in full detail and at a leisurely pace. For the first half of this work, only the standard notions of an undergraduate number theory course are required. For the second half, some knowledge of the geometry of numbers is helpful.