The exceptional set for Diophantine inequality with mixed powers of primes

Abstract: Assume that $λ_1, λ_2, λ_3,λ_4,λ_5,λ_6,λ_7$ are non-zero real numbers , $λ_1/λ_2$ is an irrational number. Let $\mathcal{V} $ be a well-spaced sequence, and $δ>0$. For any given positive integer $k\geq 5$ and any $\varepsilon >0$, we give the upper bound of the number of $\upsilon \in \mathcal{V} $ with $\upsilon \leq X$ for which the inequality $$ \left | λ_1p_1^2 + λ_2p_2^3 + λ_3p_3^3 + λ_4p_4^3 + λ_5p_5^3 + λ_6p_6^4 + λ_7p_7^k - \upsilon \right | <{\upsilon}^{-δ} $$ has no solution in primes $p_1, p_2, p_3, p_4, p_5, p_6, p_7$.

• The paper introduces a refined hybrid circle-sieve method to bound exceptional sets in a mixed-prime power Diophantine inequality.
• It employs optimal Hölder's inequality and Fourier kernel adaptations to derive nontrivial sparsity bounds on the exceptional targets.
• The results demonstrate that for generic coefficients and k ≥ 5, nearly every well-spaced value admits a prime solution, ensuring robust analytic conclusions.

Analysis of the Exceptional Set for a Diophantine Inequality with Mixed Powers of Primes

Introduction

The paper "The exceptional set for Diophantine inequality with mixed powers of primes" (2604.22147) investigates the density and structure of integers (or real numbers in a well-spaced sequence) for which a specific Diophantine inequality involving mixed exponent primes does not admit a solution. The primary inequality under investigation is:

$$\left | \lambda_1p_1^2+\lambda_2p_2^3+\lambda_3p_3^3+\lambda_4p_4^3+\lambda_5p_5^3+\lambda_6p_6^4+\lambda_7p_7^k-\upsilon \right | < \upsilon^{-\delta}$$

with coefficients $\lambda_i$ (not all negative; $\lambda_1/\lambda_2$ irrational/algebraic), variable exponent $k \geq 5$, and primes $p_1$ through $p_7$, where $\upsilon$ lies in a well-spaced sequence $\mathcal{V}$ such that $0 < c < \upsilon_{i+1} - \upsilon_i < C$. The research focuses on establishing upper bounds on the exceptional set—those $\upsilon \leq X$ with no prime solution to the inequality.

Context and Prior Work

This topic is situated at the intersection of analytic number theory, additive prime problems, and Diophantine approximation. Previous foundational works (Yu, Li, Xi & Mu) have shown the existence of infinitely many integer solutions for related inequalities with mixed powers, employing combinations of the circle and sieve methods to handle additive prime variables. The present work distinguishes itself by focusing explicitly on exceptional sets—sets of targets for which the inequality fails irrespective of the choice of primes—under more general power and coefficient restrictions and for sequences with specified spacing properties.

Main Theorem and Its Significance

The paper's principal theorem quantifies the exceptional set's size:

Let $\lambda_i$0, $\lambda_i$1 a well-spaced sequence, $\lambda_i$2 irrational and algebraic, and $\lambda_i$3. Then for any $\lambda_i$4, the number $\lambda_i$5 of $\lambda_i$6 with $\lambda_i$7 having no prime solution is

$\lambda_i$8

where

$\lambda_i$9

and $\lambda_1/\lambda_2$0 is explicitly defined per $\lambda_1/\lambda_2$1.

This bound is nontrivial, improving on prior results by integrating the primes, the mixed powers, and the well-spaced sequence structure. The theorem asserts that the set of exceptions is sparse for large $\lambda_1/\lambda_2$2, given sufficiently generic coefficients and power combinations.

Methodological Innovations

The research advances the analytic framework by:

• Circle Method with Sieve Augmentation: The paper employs a sophisticated hybrid of the Davenport-Heilbronn circle method and sieve theory, allowing high precision estimation for the major, minor, and trivial arcs.
• Optimal Use of Hölder's Inequality: Novel applications of Hölder’s inequality lead to sharper integral estimates, crucial for bounding contributions from minor/trivial arcs.
• Fourier Kernel Adaptations: The kernel $\lambda_1/\lambda_2$3 is carefully designed to relate proximity of Diophantine expressions to $\lambda_1/\lambda_2$4, permitting translation from additive prime power sums to integrals over arc segments.
• Explicit Dyadic Decomposition and Rational Approximations: The density of exceptional values is controlled by decomposing arc sets and leveraging rational approximations (via convergents to irrational/algebraic ratios of coefficients).

Strong Numerical Results and Contradictory Claims

The theorem provides explicit exponents for the upper bound on the exceptional set, parameterized via $\lambda_1/\lambda_2$5 and structured via the formula for $\lambda_1/\lambda_2$6, revealing precise sparsity for the exceptional $\lambda_1/\lambda_2$7. The bound improves substantially over previous estimates, emphasizing that for generic parameters (proper irrationality and algebraicity, $\lambda_1/\lambda_2$8), almost all $\lambda_1/\lambda_2$9 (as $k \geq 5$0) admit a solution in primes—apart from a quantitatively negligible subset.

One bold theoretical claim is that the sparsity bound persists even when the coefficients are only required to be algebraic and not transcendental—demonstrating that strong Diophantine approximation properties suffice.

Implications and Prospects for Further Research

From a practical perspective, the result underscores the robustness of circle and sieve methods for additive problems involving primes, suggesting further avenues for quantitative density results in mixed-power prime representations. The techniques may generalize to wider families of Diophantine inequalities, including those with more variables or different power mixtures, as well as to higher-dimensional exceptional sets.

Theoretically, the explicit upper bounds lay a foundation for potential refinements using improved exponential sum estimates or deeper sieve constructions. The structure of exceptional sets for additive problems with primes remains a fertile area, with possible connections to randomness in additive representations and computational complexity of searching for solutions.

Anticipated future developments may include:

• Extension to more general coefficient spaces (beyond algebraic/irrational): e.g., transcendental ratios.
• Investigation into lower bounds and explicit characterization for exceptional sets.
• Application to related additive prime problems, including Waring-Goldbach-type systems with mixed exponents.

Conclusion

The paper delivers a rigorous and technically intricate upper bound for exceptional sets in a Diophantine inequality involving mixed powers of primes, using advanced analytic and combinatorial tools. The outlined methodology not only advances the theory but also establishes effective density results, with compelling implications for additive number theory and Diophantine approximation. The work provides a basis for further refinements in exceptional set theory for prime-powered additive equations and inequalities.