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Diophantine inequalities. (English) Zbl 0592.10029
London Mathematical Society Monographs. New Series, 1. Oxford: Clarendon Press. XII, 275 p. Ł 32.00 (1986).
Based on different principal methods and types of approximation, the contents of this book can be grouped into two subjects, namely, (i) bounds for values of polynomials and forms modulo one and (ii) solubility of equations and inequalities on forms. H. Davenport and H. Heilbronn are two of the most influential pioneers in the development of these fields. They had made essential progress in the late of forties. In recent years W. M. Schmidt has been making significant contributions in both methods and results to these fields. For example, his lattice method [Small fractional parts of polynomials (Reg. Conf. Ser. Math. 32) (1977; Zbl 0362.10032)] has been found to be powerful in improving simultaneous bounds for polynomials and his investigations on small solutions of additive equations and congruences enable him to settle a long lasting conjecture on the real form of odd degree [Adv. Math. 38, 128-151 (1980; Zbl 0448.10019)]. These breakthroughs will certainly have profound influence to future investigations in these fields in many years to come. So the present book appears at right time.
Concerning the above (i) the classical Dirichlet’s theorem on approximation says that for any real \(\alpha\) and any \(N\in {\mathbb{N}}\) there exists a positive integer \(n\leq N\) such that \(\| \alpha n\| <N^{-1}\) where, for real t, \(\| t\|\) denotes \(\min_{m\in {\mathbb{Z}}}| t-m|.\) Improving a result of I. M. Vinogradov, H. Heilbronn obtained [Q. J. Math., Oxf. Ser. 19, 249-256 (1948; Zbl 0031.20502)] that for \(\alpha\),N as above and for any \(\epsilon >0\) there exist \(n\in {\mathbb{N}}\) and a constant \(C_ 1(\epsilon)>0\) such that \(n\leq N\) and \(\| \alpha n^ 2\| <C_ 1 N^{\epsilon -1/2}.\) Since then no essential improvement on the exponent \(\epsilon\)-1/2 has been obtained. However, Heilbronn’s paper and later H. Davenport’s work on this problem [Q. J. Math., Oxf. II. Ser. 18, 339-344 (1967; Zbl 0155.377)] stimulated a series of investigations in other directions. In Chapters 2, 3 and 5 of the present book generalizations of the above Heilbronn’s result to monomials and polynomials i.e. \(\| \alpha n^ k\|\) and \(\| f(n)\|\), where \(f(n)=\alpha_ kn^ k+...+\alpha_ 1n)\) are discussed. In Chapters 4 and 6, when k is large, better bounds for the above fractional parts are obtained by means of Vinogradov’s mean value method. Schmidt’s lattice method is introduced and its applications to simultaneous inequalities for polynomials (i.e. \(\| f_ j(n)\| <N^{\epsilon -c(k,h)}\) and \(n\leq N\), where N is large and \(j=1,...,h)\) are presented in Chapters 7 and 8. Extensions of Heilbronn’s result to simultaneous inequalities for real quadratic forms and real diagonal forms are given in Chapters 9 and 10.
Concerning the above (ii) a well-known theorem, due to Meyer in 1884, tells that an indefinite quadratic form in five variables with integral coefficients presents zero non-trivially. In 1957, B. J. Birch [Mathematika 4, 102-105 (1957; Zbl 0081.045)] made remarkable progress and proved that if \(G({\mathfrak x})=G(x_ 1,...,x_ s)\) is a form of odd degree k with integral coefficients and if \(s\geq C_ 2(k)\) then there is an integer point \({\mathfrak x}\neq 0\) such that G(\({\mathfrak x})=0\). For real forms it was once an old conjecture that if \(F({\mathfrak x})=F(x_ 1,...,x_ s)\) is a form of odd degree k with real coefficients and if \(s\geq C_ 3(k)\) then there is an integer point \({\mathfrak x}\neq 0\) such that \(| F({\mathfrak x})| <1.\) About certain special diagonal real forms of any degree, H. Davenport and H. Heilbronn [J. Lond. Math. Soc. 21, 185-193 (1946; Zbl 0060.119)] affirmed the above conjecture and gave \(C_ 3=2^ k+1\). In their paper they used a variant of the Hardy-Littlewood circle method called by the author in the present book as the Davenport-Heilbronn circle method. In the late seventies, Schmidt published two important papers on small solutions of diagonal equations with integral coefficients and finally in 1980 he succeeded in proving the above long lasting conjecture on the real forms of odd degree by combination of the Davenport-Heilbronn circle method and Birch’s method. He also extended his results to systems of real forms of odd degree. All these exciting results due to Schmidt are proved in Chapters 11 to 14 of the present book. In Chapters 15 to 18, the author considers the latest results concerning systems of integral forms with even degree. These four chapters consist of results given in three of the most recent papers of Schmidt on certain exponential sums of integral forms and of polynomials having coefficients in some finite fields, and their applications to small solutions of simultaneous congruences of integral forms with even degree.
In the book the author has collected most of the recent results on small fractional parts of polynomials and forms, and Schmidt’s recent work on forms and diagonal equations. The author gives an overall account of the present state of knowledge on the above subjects and also supplies information on historical developments of these subjects from Davenport- Heilbronn’s work to Schmidt’s one. The book is carefully written, readable and is a good reference for graduate students who are interested in Diophantine approximation and related topics although the author gives no space in this book to prime solutions of Diophantine inequalities and equations. The book may be read as a supplement to Schmidt’s book published in 1977 (loc. cit.).
[Reviewer: Schmidt’s results discussed in Chapters 11 to 14 of the present book have been extended to algebraic number fields by Yuan Wang (to appear in Acta Arith.; see the previews in Zbl 0569.10007 and Zbl 0576.10013)].
Reviewer: M.-C.Liu

11J71 Distribution modulo one
11P55 Applications of the Hardy-Littlewood method
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11E16 General binary quadratic forms
11D72 Diophantine equations in many variables
11D75 Diophantine inequalities