The Sharp Sadov Constant and Local Spectral Stability for Shapiro--Diananda Cyclic Sums

Abstract: We determine the sharp Sadov constant for Shapiro--Diananda cyclic sums. Sadov proved the lower bound C >= log 2; we prove the matching upper bound by an explicit asymptotic construction, obtaining C = log 2. We also develop a local spectral stability theory for the equal point of the Shapiro--Diananda cyclic sums. The Hessian is diagonalized by Fourier modes, giving an exact local minimum/saddle/quadratic-degeneracy criterion for all n and k, periodic equality families, and explicit classifications for k = 2 and k = 3. The result determines the global infimum over all n and k, but does not solve the separate fixed-k asymptotic minimization problems.

• The paper establishes the exact value of the global Sadov constant as log 2 through an asymptotic construction.
• It uses Fourier diagonalization to analyze the circulant quadratic form, detailing stability, degeneracy, and saddle behaviors near the symmetric point.
• The study fully resolves the cases for k=2 and k=3 while providing criteria for local stability in larger cyclic sums.

The Sharp Sadov Constant and Local Spectral Stability for Shapiro–Diananda Cyclic Sums

Overview and Motivation

This work provides a comprehensive analysis of the Shapiro–Diananda cyclic sums,

$$S_{n,k}(x) = \sum_{i=1}^n \frac{x_i}{x_{i+1}+\cdots+x_{i+k}}, \quad x_i>0,$$

with cyclic indices. The focus is twofold: (1) a precise characterization of the local quadratic behavior near the symmetric point $x_1 = \cdots = x_n$, and (2) the identification of the global sharp constant ("Sadov constant") for normalized sums, a longstanding extremal problem in cyclic inequalities.

The study builds on and extends the program initiated by Sadov and others, providing both new local spectral results and resolving the global constant exactly.

Local Spectral Theory: Quadratic Expansion, Fourier Analysis, and Stability Trichotomy

Logarithmic Parameterization and Quadratic Expansion

Exploiting the homogeneity of $S_{n,k}$, the variables are transformed via $x_i = e^{u_i}$ with $\sum_{i=1}^n u_i = 0$, focusing on perturbations in the log-coordinates orthogonal to scaling. The Taylor expansion of $S_{n,k}$ near the symmetric point yields:

$$S_{n,k}(e^u) = \frac{n}{k} + Q_{n,k}(u) + O(\|u\|^3),$$

where $Q_{n,k}$ is a circulant quadratic form.

Circulant Hessian and Fourier Diagonalization

$Q_{n,k}$ is diagonalized by the discrete Fourier basis, leading to explicit expressions for its eigenvalues:

$$Q_{n,k}(u) = \sum_{m=1}^{n-1} \gamma_{m,k} |\widehat{u}_m|^2.$$

The core formula for the symbol is:

$x_1 = \cdots = x_n$0

Sharp Local Constant and Trichotomy

Define:

$x_1 = \cdots = x_n$1

There is a sharp trichotomy:

• $x_1 = \cdots = x_n$2: the symmetric point is a strict local minimum; local quadratic stability holds.
• $x_1 = \cdots = x_n$3: quadratic degeneracy (zero second-order term along some direction).
• $x_1 = \cdots = x_n$4: the symmetric point is a saddle; $x_1 = \cdots = x_n$5 arbitrarily close to equality.

Explicit Classification for $x_1 = \cdots = x_n$6 and $x_1 = \cdots = x_n$7

The cases $x_1 = \cdots = x_n$8 and $x_1 = \cdots = x_n$9 are fully resolved.

$S_{n,k}$0:

$S_{n,k}$1 for odd $S_{n,k}$2 (strict minimum), and is $S_{n,k}$3 for even $S_{n,k}$4 (degeneracy). Thus, local stability only arises for odd cycles.

$S_{n,k}$5:

There is a finite, completely classified set of $S_{n,k}$6 such that $S_{n,k}$7:

$S_{n,k}$8

For $S_{n,k}$9 divisible by $x_i = e^{u_i}$0 but not one of these, degeneracy arises; for all other $x_i = e^{u_i}$1, the point is a saddle. As $x_i = e^{u_i}$2 increases, local instability emerges for fixed $x_i = e^{u_i}$3.

Periodic Equality Families and Obstructions

A general mechanism is identified: when $x_i = e^{u_i}$4, $x_i = e^{u_i}$5-periodic families attain $x_i = e^{u_i}$6, producing global nonconstant equality cases and obstructing strict local stability. This analysis explains spectral degeneracies as consequences of periodic families.

Uniform and Quantitative Saddle Criteria

For each fixed $x_i = e^{u_i}$7, the work shows that the symmetric point is a saddle for all sufficiently large $x_i = e^{u_i}$8, using a detailed analysis of the Fourier symbol. A simple sufficient condition is $x_i = e^{u_i}$9, though sharper computable bounds are provided via analysis of the first negative interval of the symbol $\sum_{i=1}^n u_i = 0$0. For small values of $\sum_{i=1}^n u_i = 0$1, the threshold is often significantly below $\sum_{i=1}^n u_i = 0$2.

The Sharp Global Sadov Constant

Problem Statement

The global problem seeks

$\sum_{i=1}^n u_i = 0$3

with the "Sadov constant" $\sum_{i=1}^n u_i = 0$4 representing the minimal normalized sum over all admissible $\sum_{i=1}^n u_i = 0$5.

Lower and Upper Bounds

Sadov had previously established the lower bound $\sum_{i=1}^n u_i = 0$6. The present work provides an explicit asymptotic construction achieving the upper bound, thereby confirming the exact value:

$\sum_{i=1}^n u_i = 0$7

Asymptotic Construction

The upper bound is achieved via a configuration consisting of two identical sharply concentrated periods of weights and a small additional window; as $\sum_{i=1}^n u_i = 0$8, the limiting behavior of the normalized sum is shown to yield exactly $\sum_{i=1}^n u_i = 0$9. The argument utilizes careful periodic concentration estimates and a telescoping sum analogous to a discrete logarithmic integral.

Implications and Prospective Developments

The established value $S_{n,k}$0 provides a definitive answer to the global normalization problem for Shapiro–Diananda sums, resolving a longstanding question and setting a reference for future work on related cyclic and graphic inequalities.

On the local side, the spectral framework rigorously characterizes stability for all $S_{n,k}$1, fully resolving the quadratic behavior for $S_{n,k}$2 and presenting sharp criteria for general $S_{n,k}$3. The arithmetic sensitivity and the emergence of instability in large cycles ($S_{n,k}$4) are of particular note.

The interplay between local (Hessian) and global (concentration) mechanisms clarifies how extremal configurations evade symmetric points, especially as $S_{n,k}$5 becomes large. The analysis suggests further investigation into the asymptotic behavior of fixed-$S_{n,k}$6 constants and possible extensions to weighted or non-cyclic variants, as well as connections to spectral graph theory and circulant quadratic form analysis.

Conclusion

This paper delivers a complete spectral-local classification for the stability of Shapiro–Diananda cyclic sums and resolves the sharp normalized global lower bound as $S_{n,k}$7. The results elucidate the complex arithmetic and spectral structure governing cyclic inequalities, bridging local stability, global extremality, and periodicity phenomena within a unified analytic framework. These findings provide a solid foundation for continued exploration of cyclic, graphic, and spectral inequalities in analysis and combinatorics.