Generalized Estermann problem for non-integer powers with almost proportional summands

Abstract: For $H \ge N^{1-\frac{1}{2c}} \ln^2 N$, where $c$ is a fixed non-integer number satisfying $$ |c| \ge 3c\left(2^{[c]+1}-1\right)\frac{\ln \ln N}{\ln N}, \qquad c > \frac{4}{3}\left(1 + \frac{52\ln \ln N}{\ln N}\right), $$ we obtain an asymptotic formula for the number of representations of a sufficiently large integer $N$ in the form $$ p_{1} + p_{2} + [n^{c}] = N, $$ where $p_{1}, p_{2}$ are prime numbers, $n$ is a natural number, and $$ |p_{k} - μ{k}N| \le H,\qquad k = 1,2,\qquad |[n^{c}] - μ{3}N| \le H, $$ with $μ{1}, μ{2}, μ{3}$ being fixed positive constants satisfying $μ{1} + μ{2} + μ{3} = 1$. Keywords: Estermann problem, almost proportional summands, short exponential sum with a non-integer power of a natural number. Bibliography: 21 references.

• The paper extends the classical Estermann problem by analyzing non-integer powers with summands concentrated near specified proportions.
• It employs a modified Hardy-Littlewood circle method with refined exponential sum estimates to overcome challenges in short interval analysis.
• The results bridge a key gap in additive number theory by providing explicit asymptotic formulas under Diophantine conditions for the exponent.

Generalized Estermann Problem for Non-Integer Powers and Almost Proportional Summands

Introduction and Problem Statement

The paper addresses a significant generalization of the classical additive Estermann problem, extending its study to Diophantine equations of the form $p_1 + p_2 + [n^c] = N$, where $p_1$ and $p_2$ are primes, $n$ is a natural number, and $c > 1$ is a fixed non-integer. The focus is on the enumeration of representations where each summand is constrained to lie within a short interval centered around a fixed positive proportion of $N$. Specifically, for real parameters $\mu_1, \mu_2, \mu_3 > 0$ with $\mu_1 + \mu_2 + \mu_3 = 1$, and for a short interval of size $H \geq N^{1 - \frac{1}{2c}} L^2$ (with $L = \ln N$), the paper establishes an explicit asymptotic formula for the count $p_1$0 of solutions satisfying:

$p_1$1

This extends prior work, which mainly considered integer powers and equi-distributed or almost equal summands, to the analytically challenging regime of non-integer exponents and proportionally distributed variables.

Background and Context

The Estermann problem, originating with T. Estermann, concerns additive decompositions of large integers as sums of primes and powers, most notably in the equation $p_1$2. Analytic results for the count of representations with all summands in short intervals have played a pivotal role in additive number theory, with substantial progress for integer exponents and nearly-equal summands (see, e.g., [Estermann], [RakhmonovZKh-Matzametki-2014-95-3]).

A harder variant was posed by Chubarikov: replace integer powers with non-integer powers, i.e., consider $p_1$3 with $p_1$4, and establish corresponding asymptotic formulas. Substantial technical obstacles arise from the absence of simple algebraic structure and the analytic behavior of exponential sums $p_1$5 in this setting, particularly for short intervals and in the presence of Diophantine discrepancies in $p_1$6. Previous work solved the "almost equal summands" case [RakhmonovPZ-MZ-2016]; the present paper advances the theory to "almost proportional summands," completing an important piece of the generalized program.

Main Theorem and Key Technical Claims

For $p_1$7 sufficiently large, $p_1$8, and with Diophantine constraints on $p_1$9 given by

$p_2$0

the paper proves the following asymptotic formula for the count $p_2$1 as defined above: $p_2$2 provided $p_2$3.

This formula generalizes previous results ($p_2$4) and significantly relaxes the constraints on the distribution of the summands, allowing them to be "almost proportional," that is, concentrated near arbitrary positive proportions of $p_2$5.

Analytical Methods and Innovations

The core technique is the Hardy-Littlewood circle method, modified to accommodate non-integer powers and short intervals. Several critical innovations support the main result:

• Refined Estimates for Short Exponential Sums: The authors employ recent, sharp bounds for $p_2$6, uniform in $p_2$7, leveraging intricate properties of Weyl-type sums and their decomposition over major and minor arcs (see [RakhmonovPZ-MZ-2014]).
• Zero Density Estimates for the Riemann Zeta Function: The precise control on error terms relies on density theorems for zeros of $p_2$8 in the critical strip, employing contemporary bounds from [Zhan Tao-Acta-Math-Sinica-1992].
• Diophantine Conditions on the Exponent $p_2$9: The analytic behavior of $n$0 in exponential sums is sensitive to the Diophantine approximation property of $n$1; the authors mandate explicit lower bounds for $n$2 to control pseudo-randomness and preclude resonance effects.
• Saddle Point Analysis and Truncated Fourier Integrals: For the main term computation, the transformation of discretized interval counts to integrals allows asymptotic evaluation, with meticulous handling of remainder terms.

Compared to previous studies—for example, [RakhmonovPZ-MZ-2016]'s results for almost equal summands—the present work not only accommodates general proportions but also achieves sharper error terms and covers a broader class of exponents.

Numerical Strength and Contradictory Claims

The principal numerical claim is the sharp $n$3 leading order in the numerator and an explicit dependence on $n$4 in the denominator, which reflects the interplay between the scaling of the almost proportional window and the analytic density of suitable representations. The main term's coefficient matches straightforward probabilistic heuristics, but the error term's power savings $n$5 advances prior art. No contradictory claims are made relative to established results for integer powers or equal summands; instead, the new regime is handled distinctly.

Theoretical and Practical Implications

The extension to non-integer exponents and proportional localization of summands closes a long-standing gap in the Estermann problem and its multiplicative generalizations. The result achieves, for the first time, an analytic description of representations in this general setting, which is crucial for understanding the fine structure of additive representation functions in sparse sequences (e.g., "fractional" powers).

Practically, this framework strengthens heuristic models in computational additive number theory, showing that "fractional" structures admit the same kind of universality as classical Waring and Goldbach-type problems under appropriate hypotheses.

Theoretically, the methods underscore the necessity of sophisticated exponential sum theory and zeta-density bounds for results dealing with non-integer polynomial phases in analytic number theory. The proof illustrates the delicate sensitivity of such questions to Diophantine properties of the parameters, a theme likely central in future work on additive decompositions with transcendental or non-algebraic components.

Speculation on Future Directions

Future developments may include:

• Extension to more general additive equations with several primes and arbitrary non-integer powers.
• Sharper and more general bounds as exponential sum technology and zero-density estimates progress.
• Deepening the use of metric Diophantine approximation to relax or quantify the restrictions on $n$6.
• Connections with additive combinatorics and pseudo-randomness in additive sequences, possibly informing models in probabilistic number theory.

Conclusion

The paper systematically resolves the generalized Estermann problem for non-integer exponents with almost proportional summands, obtaining an explicit asymptotic formula for the number of representations of sufficiently large integers as the sum of two primes and a non-integer power in prescribed intervals. The depth of analytic machinery and the generality of the proportional window establish a strong foundation for subsequent work in additive number theory and its intersections with analytic and Diophantine analysis.