A simple but effective upper bound technique is to have a quantity expressed by a Cauchy integral, choose an optimal contour $\gamma$, and proceed by inserting the absolute values along $\gamma$. For example, a clean estimate $\log{\binom{a}{b}} \leq b \log(b/a) + (a-b)\log(1-b/a)$ on the binomial coefficients follows from expressing $$ \binom{a}{b} = \int_{|z| = R} \frac{(1-z)^a}{z^b} \, d\theta \leq R^{-b} (1+R)^a $$$$ \binom{a}{b} = \Big| \int_{|z| = R} \frac{(1-z)^a}{z^b} \, d\theta \Big| \leq R^{-b} (1+R)^a $$ and noting that the bound is optimized for the choice $R = b/(a-b)$. This is the same idea as in Lucia's solution to your problem, but with the additionaladded point that the contour could be varied and a good choice of it is part of the techniquecontour is on disposal. Many papers on diophantine approximations involve this idea in extrapolation arguments, e.g. Gelfond's solution to Hilbert's 7th problem.
A simple but effective upper bound technique is to have a quantity expressed by a Cauchy integral, choose an optimal contour $\gamma$, and proceed by inserting the absolute values along $\gamma$. For example, a clean estimate $\log{\binom{a}{b}} \leq b \log(b/a) + (a-b)\log(1-b/a)$ on the binomial coefficients follows from expressing $$ \binom{a}{b} = \int_{|z| = R} \frac{(1-z)^a}{z^b} \, d\theta \leq R^{-b} (1+R)^a $$ and noting that the bound is optimized for the choice $R = b/(a-b)$. This is the same idea as in Lucia's solution to your problem, but with the additional point that the contour could be varied and a good choice of it is part of the technique. Many papers on diophantine approximations involve this idea in extrapolation arguments, e.g. Gelfond's solution to Hilbert's 7th problem.
A simple but effective upper bound technique is to have a quantity expressed by a Cauchy integral, choose an optimal contour $\gamma$, and proceed by inserting the absolute values along $\gamma$. For example, a clean estimate $\log{\binom{a}{b}} \leq b \log(b/a) + (a-b)\log(1-b/a)$ on the binomial coefficients follows from expressing $$ \binom{a}{b} = \Big| \int_{|z| = R} \frac{(1-z)^a}{z^b} \, d\theta \Big| \leq R^{-b} (1+R)^a $$ and noting that the bound is optimized for the choice $R = b/(a-b)$. This is the same idea as in Lucia's solution to your problem, with the added point that the choice of the contour is on disposal. Many papers on diophantine approximations involve this idea in extrapolation arguments, e.g. Gelfond's solution to Hilbert's 7th problem.
A simple but effective upper bound technique is to have a quantity expressed by a Cauchy integral, choose an optimal contour $\gamma$, and proceed by inserting the absolute values along $\gamma$. For example, a clean estimate $\log{\binom{a}{b}} \leq b \log(b/a) + (a-b)\log(1-b/a)$ on the binomial coefficients follows from expressing $$ \binom{a}{b} = \int_{|z| = R} \frac{(1-z)^a}{z^b} \, d\theta \leq R^{-b} (1+R)^a $$ and noting that the bound is optimized for the choice $R = b/(a-b)$. This is the same idea as in Lucia's solution to your problem, but with the additional point that the contour could be varied and a good choice of it is part of the technique.
Many papers on diophantine approximations involve this idea in extrapolation arguments. It is everywhere in A. O. Gel'fond's papers, themselves inspired by work of Polya, Hardy, and Fukasawa on integer-valued entire functions with slow growth, that solved the Hilbert 7th problem and, going further than this, obtained explicit estimates for linear forms in two logarithms of algebraic numbers. Retrospectively, to name perhaps the most important application, Gel'fond's estimates have turned out to be sufficient for the solution of Gauss's class number one problem, as Stark explains in the two-page paper [A historical note on complex quadratic fields with class number one, Proc. Amer. Math. Soc., 1969]. Alan Baker's extension to linear forms in more than two logarithms, while involving a more elaborate extrapolation procedure, was also based on the same idea with the Cauchy integral.
Nowadays the viewpoint, in this and similar topics from transcendence theory, is usually different. The extrapolation methodology is discarded altogether. Instead, one constructs (using a zero estimate) a non-vanishing ``interpolation determinant,'' like $\det(f_i(\zeta_j))_{i,j}$ or a similar multivariable alterant involving derivatives. Here, $f_i$ are certain analytic functions, while $\zeta_j$ are the (usually) algebraic points of interest. Under appropriate assumptions one has a diophantine lower bound on such a determinant, whereas an upper bound is given by Schwarz's lemma, since for example $z \mapsto \det(f_i(\zeta_j z))_{i,j=1}^L$ vanishes at $z = 0$ with multiplicity $\binom{L}{2}$. Comparing the two bounds then produces a diophantine estimate, a transcendence or, when viewed contrapositively, an algebraicity result. A comprehensive exposition, and comparison, of both the extrapolation and alterant methods in transcendence is given in Waldschmidt's book Diophantine Approximation on Linear Algebraic Groups.
This latter method was discovered independently by M. Laurent [Sur quelques resultats recents de transcendence, 1989] and Je. Pila [Geometric and arithmetic postulation of the exponential function, 1993]g. The key point is now the analyticity of the functions $f_i$ and not the Cauchy integral; indeed the determinental method has been used by Bombieri and PilaGelfond's solution to bound the number of integer points lying on a real analytic, but not algebraic, arc [The number of integral points on arcs and ovals, 1989]. This has stimulated a lot of developments in number theory, diophantine geometry, and logic, but at this point I veer away from the question.
I will end by remarking that the same idea of using a rational approximation and a Schwarz bound has been applied in complex geometry also. In its most primitive form it was used by Siegel in [Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten, 1955] to reprove the bound (Thimm's theorem) $\mathrm{tr.deg.}_{\mathbb{C}} \mathcal{M}(X) \leq \dim_{\mathbb{C}}{X}$ on the field of meromorphic functions on a compact complex manifold $X$. This is a basic foundational result that includes as a special case Chow's theorem on the algebraicity of projective analytic sets. An advanced form of the same technique is, for example, one of the components in Siu's proof of the Grauert-Riemenschneider conjecture on Moishezon spaces and semipositive line bundles [A vanishing theorem for semipositive line bundles over non-Kahler manifolds, 1984]Hilbert's 7th problem.
A simple but effective upper bound technique is to have a quantity expressed by a Cauchy integral, choose an optimal contour $\gamma$, and proceed by inserting the absolute values along $\gamma$. For example, a clean estimate $\log{\binom{a}{b}} \leq b \log(b/a) + (a-b)\log(1-b/a)$ on the binomial coefficients follows from expressing $$ \binom{a}{b} = \int_{|z| = R} \frac{(1-z)^a}{z^b} \, d\theta \leq R^{-b} (1+R)^a $$ and noting that the bound is optimized for the choice $R = b/(a-b)$. This is the same idea as in Lucia's solution to your problem, with the additional point that the contour could be varied and a good choice of it is part of the technique.
Many papers on diophantine approximations involve this idea in extrapolation arguments. It is everywhere in A. O. Gel'fond's papers, themselves inspired by work of Polya, Hardy, and Fukasawa on integer-valued entire functions with slow growth, that solved the Hilbert 7th problem and, going further than this, obtained explicit estimates for linear forms in two logarithms of algebraic numbers. Retrospectively, to name perhaps the most important application, Gel'fond's estimates have turned out to be sufficient for the solution of Gauss's class number one problem, as Stark explains in the two-page paper [A historical note on complex quadratic fields with class number one, Proc. Amer. Math. Soc., 1969]. Alan Baker's extension to linear forms in more than two logarithms, while involving a more elaborate extrapolation procedure, was also based on the same idea with the Cauchy integral.
Nowadays the viewpoint, in this and similar topics from transcendence theory, is usually different. The extrapolation methodology is discarded altogether. Instead, one constructs (using a zero estimate) a non-vanishing ``interpolation determinant,'' like $\det(f_i(\zeta_j))_{i,j}$ or a similar multivariable alterant involving derivatives. Here, $f_i$ are certain analytic functions, while $\zeta_j$ are the (usually) algebraic points of interest. Under appropriate assumptions one has a diophantine lower bound on such a determinant, whereas an upper bound is given by Schwarz's lemma, since for example $z \mapsto \det(f_i(\zeta_j z))_{i,j=1}^L$ vanishes at $z = 0$ with multiplicity $\binom{L}{2}$. Comparing the two bounds then produces a diophantine estimate, a transcendence or, when viewed contrapositively, an algebraicity result. A comprehensive exposition, and comparison, of both the extrapolation and alterant methods in transcendence is given in Waldschmidt's book Diophantine Approximation on Linear Algebraic Groups.
This latter method was discovered independently by M. Laurent [Sur quelques resultats recents de transcendence, 1989] and J. Pila [Geometric and arithmetic postulation of the exponential function, 1993]. The key point is now the analyticity of the functions $f_i$ and not the Cauchy integral; indeed the determinental method has been used by Bombieri and Pila to bound the number of integer points lying on a real analytic, but not algebraic, arc [The number of integral points on arcs and ovals, 1989]. This has stimulated a lot of developments in number theory, diophantine geometry, and logic, but at this point I veer away from the question.
I will end by remarking that the same idea of using a rational approximation and a Schwarz bound has been applied in complex geometry also. In its most primitive form it was used by Siegel in [Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten, 1955] to reprove the bound (Thimm's theorem) $\mathrm{tr.deg.}_{\mathbb{C}} \mathcal{M}(X) \leq \dim_{\mathbb{C}}{X}$ on the field of meromorphic functions on a compact complex manifold $X$. This is a basic foundational result that includes as a special case Chow's theorem on the algebraicity of projective analytic sets. An advanced form of the same technique is, for example, one of the components in Siu's proof of the Grauert-Riemenschneider conjecture on Moishezon spaces and semipositive line bundles [A vanishing theorem for semipositive line bundles over non-Kahler manifolds, 1984].
A simple but effective upper bound technique is to have a quantity expressed by a Cauchy integral, choose an optimal contour $\gamma$, and proceed by inserting the absolute values along $\gamma$. For example, a clean estimate $\log{\binom{a}{b}} \leq b \log(b/a) + (a-b)\log(1-b/a)$ on the binomial coefficients follows from expressing $$ \binom{a}{b} = \int_{|z| = R} \frac{(1-z)^a}{z^b} \, d\theta \leq R^{-b} (1+R)^a $$ and noting that the bound is optimized for the choice $R = b/(a-b)$. This is the same idea as in Lucia's solution to your problem, but with the additional point that the contour could be varied and a good choice of it is part of the technique. Many papers on diophantine approximations involve this idea in extrapolation arguments, e.g. Gelfond's solution to Hilbert's 7th problem.
A simple but effective upper bound technique is to have a quantity expressed by a Cauchy integral, choose an optimal contour $\gamma$, and proceed by inserting the absolute values along $\gamma$. For example, a clean estimate $\log{\binom{a}{b}} \leq b \log(b/a) + (a-b)\log(1-b/a)$ on the binomial coefficients follows from expressing $$ \binom{a}{b} = \int_{|z| = R} \frac{(1-z)^a}{z^b} \, d\theta \leq R^{-b} (1+R)^a $$ and noting that the bound is optimized for the choice $R = b/(a-b)$. This is the same idea as in Lucia's solution to your problem, with the additional point that the contour could be varied and a good choice of it is part of the technique.
All the earlyMany papers on effective diophantine approximations usedinvolve this type of idea in extrapolation arguments. It is everywhere in A. O. Gel'fond's papers, themselves inspired by work of Polya, Hardy, and Fukasawa on integer-valued entire functions with slow growth, that solved the Hilbert 7th problem and, going further than this, obtained explicit estimates for linear forms in two logarithms of algebraic numbers. Retrospectively, to name perhaps the most important application, Gel'fond's estimates have turned out to be sufficient for the solution of Gauss's class number one problem, as Stark explains in the two-page paper [A historical note on complex quadratic fields with class number one, Proc. Amer. Math. Soc., 1969]. Alan Baker's extension to linear forms in more than two logarithms, while involving a more elaborate extrapolation procedure, was also based on the same idea with the Cauchy integral.
Nowadays the viewpoint, in this and similar topics from transcendence theory, is usually different. The extrapolation methodology is discarded altogether. Instead, one constructs (using a zero estimate) a non-vanishing ``interpolation determinant,'' like $\det(f_i(\zeta_j))_{i,j}$ or a similar multivariantmultivariable alterant involving derivatives. Here, $f_i$ are certain analytic functions, while $\zeta_j$ are the (usually) algebraic points of interest. Under appropriate assumptions one has a diophantine lower bound on such a determinant, whereas an upper bound is given by Schwarz's lemma, since for example $z \mapsto \det(f_i(\zeta_j z))_{i,j=1}^L$ vanishes at $z = 0$ with multiplicity $\binom{L}{2}$. Comparing the two bounds then produces a diophantine estimate, a transcendence or, when viewed contrapositively, an algebraicity result. A comprehensive exposition, and comparison, of both the extrapolation and alterant methods in transcendence is given in Waldschmidt's book Diophantine Approximation on Linear Algebraic Groups.
This latter method was discovered independently by M. Laurent [Sur quelques resultats recents de transcendence, 1989] and J. Pila [Geometric and arithmetic postulation of the exponential function, 1993]. The key point is now the analyticity of the functions $f_i$ and not the Cauchy integral; indeed the determinental method has been used by Bombieri and Pila to bound the number of integer points lying on a real analytic, but not algebraic, arc [The number of integral points on arcs and ovals, 1989]. This has stimulated a lot of developments in number theory, diophantine geometry, and logic, but at this point I veer away from the question.
I will end by remarking that the same idea of using a rational approximation and a Schwarz bound has been applied in complex geometry also. In its most primitive form it was used by Siegel in [Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten, 1955] to reprove the bound (Thimm's theorem) $\mathrm{tr.deg.}_{\mathbb{C}} \mathcal{M}(X) \leq \dim_{\mathbb{C}}{X}$ on the field of meromorphic functions on a compact complex manifold $X$. This is a basic foundational result that includes as a special case Chow's theorem on the algebraicity of projective analytic sets. An advanced form of the same technique is, for example, one of the components in Siu's proof of the Grauert-Riemenschneider conjecture on Moishezon spaces and semipositive line bundles [A vanishing theorem for semipositive line bundles over non-Kahler manifolds, 1984].
A simple but effective upper bound technique is to have a quantity expressed by a Cauchy integral, choose an optimal contour $\gamma$, and proceed by inserting the absolute values along $\gamma$. For example, a clean estimate $\log{\binom{a}{b}} \leq b \log(b/a) + (a-b)\log(1-b/a)$ on the binomial coefficients follows from expressing $$ \binom{a}{b} = \int_{|z| = R} \frac{(1-z)^a}{z^b} \, d\theta \leq R^{-b} (1+R)^a $$ and noting that the bound is optimized for the choice $R = b/(a-b)$. This is the same idea as in Lucia's solution to your problem, with the additional point that the contour could be varied and a good choice of it is part of the technique.
All the early papers on effective diophantine approximations used this type of idea in extrapolation arguments. It is everywhere in A. O. Gel'fond's papers, themselves inspired by work of Polya, Hardy, and Fukasawa on integer-valued entire functions with slow growth, that solved the Hilbert 7th problem and, going further than this, obtained explicit estimates for linear forms in two logarithms of algebraic numbers. Retrospectively, to name perhaps the most important application, Gel'fond's estimates have turned out to be sufficient for the solution of Gauss's class number one problem, as Stark explains in the two-page paper [A historical note on complex quadratic fields with class number one, Proc. Amer. Math. Soc., 1969]. Alan Baker's extension to linear forms in more than two logarithms, while involving a more elaborate extrapolation procedure, was also based on the same idea with the Cauchy integral.
Nowadays the viewpoint, in this and similar topics from transcendence theory, is usually different. The extrapolation methodology is discarded altogether. Instead, one constructs (using a zero estimate) a non-vanishing ``interpolation determinant,'' like $\det(f_i(\zeta_j))_{i,j}$ or a similar multivariant alterant involving derivatives. Here, $f_i$ are certain analytic functions, while $\zeta_j$ are the (usually) algebraic points of interest. Under appropriate assumptions one has a diophantine lower bound on such a determinant, whereas an upper bound is given by Schwarz's lemma, since for example $z \mapsto \det(f_i(\zeta_j z))_{i,j=1}^L$ vanishes at $z = 0$ with multiplicity $\binom{L}{2}$. Comparing the two bounds then produces a diophantine estimate, a transcendence or, when viewed contrapositively, an algebraicity result.
This latter method was discovered independently by M. Laurent [Sur quelques resultats recents de transcendence, 1989] and J. Pila [Geometric and arithmetic postulation of the exponential function, 1993]. The key point is now the analyticity of the functions $f_i$ and not the Cauchy integral; indeed the determinental method has been used by Bombieri and Pila to bound the number of integer points lying on a real analytic, but not algebraic, arc [The number of integral points on arcs and ovals, 1989]. This has stimulated a lot of developments in number theory, diophantine geometry, and logic, but at this point I veer away from the question.
I will end by remarking that the same idea of using a rational approximation and a Schwarz bound has been applied in complex geometry also. In its most primitive form it was used by Siegel in [Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten, 1955] to reprove the bound (Thimm's theorem) $\mathrm{tr.deg.}_{\mathbb{C}} \mathcal{M}(X) \leq \dim_{\mathbb{C}}{X}$ on the field of meromorphic functions on a compact complex manifold $X$. This is a basic foundational result that includes as a special case Chow's theorem on the algebraicity of projective analytic sets. An advanced form of the same technique is, for example, one of the components in Siu's proof of the Grauert-Riemenschneider conjecture on Moishezon spaces and semipositive line bundles [A vanishing theorem for semipositive line bundles over non-Kahler manifolds, 1984].
A simple but effective upper bound technique is to have a quantity expressed by a Cauchy integral, choose an optimal contour $\gamma$, and proceed by inserting the absolute values along $\gamma$. For example, a clean estimate $\log{\binom{a}{b}} \leq b \log(b/a) + (a-b)\log(1-b/a)$ on the binomial coefficients follows from expressing $$ \binom{a}{b} = \int_{|z| = R} \frac{(1-z)^a}{z^b} \, d\theta \leq R^{-b} (1+R)^a $$ and noting that the bound is optimized for the choice $R = b/(a-b)$. This is the same idea as in Lucia's solution to your problem, with the additional point that the contour could be varied and a good choice of it is part of the technique.
Many papers on diophantine approximations involve this idea in extrapolation arguments. It is everywhere in A. O. Gel'fond's papers, themselves inspired by work of Polya, Hardy, and Fukasawa on integer-valued entire functions with slow growth, that solved the Hilbert 7th problem and, going further than this, obtained explicit estimates for linear forms in two logarithms of algebraic numbers. Retrospectively, to name perhaps the most important application, Gel'fond's estimates have turned out to be sufficient for the solution of Gauss's class number one problem, as Stark explains in the two-page paper [A historical note on complex quadratic fields with class number one, Proc. Amer. Math. Soc., 1969]. Alan Baker's extension to linear forms in more than two logarithms, while involving a more elaborate extrapolation procedure, was also based on the same idea with the Cauchy integral.
Nowadays the viewpoint, in this and similar topics from transcendence theory, is usually different. The extrapolation methodology is discarded altogether. Instead, one constructs (using a zero estimate) a non-vanishing ``interpolation determinant,'' like $\det(f_i(\zeta_j))_{i,j}$ or a similar multivariable alterant involving derivatives. Here, $f_i$ are certain analytic functions, while $\zeta_j$ are the (usually) algebraic points of interest. Under appropriate assumptions one has a diophantine lower bound on such a determinant, whereas an upper bound is given by Schwarz's lemma, since for example $z \mapsto \det(f_i(\zeta_j z))_{i,j=1}^L$ vanishes at $z = 0$ with multiplicity $\binom{L}{2}$. Comparing the two bounds then produces a diophantine estimate, a transcendence or, when viewed contrapositively, an algebraicity result. A comprehensive exposition, and comparison, of both the extrapolation and alterant methods in transcendence is given in Waldschmidt's book Diophantine Approximation on Linear Algebraic Groups.
This latter method was discovered independently by M. Laurent [Sur quelques resultats recents de transcendence, 1989] and J. Pila [Geometric and arithmetic postulation of the exponential function, 1993]. The key point is now the analyticity of the functions $f_i$ and not the Cauchy integral; indeed the determinental method has been used by Bombieri and Pila to bound the number of integer points lying on a real analytic, but not algebraic, arc [The number of integral points on arcs and ovals, 1989]. This has stimulated a lot of developments in number theory, diophantine geometry, and logic, but at this point I veer away from the question.
I will end by remarking that the same idea of using a rational approximation and a Schwarz bound has been applied in complex geometry also. In its most primitive form it was used by Siegel in [Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten, 1955] to reprove the bound (Thimm's theorem) $\mathrm{tr.deg.}_{\mathbb{C}} \mathcal{M}(X) \leq \dim_{\mathbb{C}}{X}$ on the field of meromorphic functions on a compact complex manifold $X$. This is a basic foundational result that includes as a special case Chow's theorem on the algebraicity of projective analytic sets. An advanced form of the same technique is, for example, one of the components in Siu's proof of the Grauert-Riemenschneider conjecture on Moishezon spaces and semipositive line bundles [A vanishing theorem for semipositive line bundles over non-Kahler manifolds, 1984].