An additive application of the resonance method

1. Problem Statement

The paper addresses the problem of establishing sharp $\Omega$-results (lower bounds on the magnitude of oscillation) for error terms in classical lattice point problems, specifically:

• Dirichlet's Divisor Problem: The error term $\Delta(x) = \sum_{n \le x} d(n) - x \log x - (2\gamma - 1)x$.
• Gauss' Circle Problem: The error term $P(x) = \sum_{n \le x} r(n) - \pi x$.
• Generalizations: The $k$-divisor problem and the mean square error of the Riemann zeta-function.

Prior to this work, the best known results for these problems were established by Soundararajan (2003). Soundararajan's approach relied on Dirichlet's simultaneous approximation theorem to construct intervals where trigonometric polynomials with positive Fourier coefficients attain large values. However, this method imposed a strict limitation: the number of terms ($M$) selected from the polynomial to maximize the lower bound could only be of order $o(\log X)$. This restriction stemmed from the error terms in the approximation theorem, which became dominant if $M$ grew too large relative to the interval length.

The central question is: Can one improve the lower bounds for these error terms by relaxing the restriction on the number of terms ($M$) used in the resonance construction?

2. Methodology: The Resonance Method as an Additive Device

The author introduces a novel application of the resonance method, previously developed by Soundararajan primarily for multiplicative problems (e.g., extreme values of the Riemann zeta-function on the critical line).

• Traditional Resonance: Typically involves a Dirichlet polynomial $R(t)$ resonating with a multiplicative Dirichlet polynomial $F(t)$.
• Additive Extension: Sourmelidis extends this to general trigonometric series $F(x) = \sum f(n)e(\lambda_n x)$ with positive Fourier coefficients.
• The Mechanism:
  1. Instead of Dirichlet's approximation, the author constructs a resonator $R(x)$ which is a trigonometric polynomial.
  2. The frequencies of $R(x)$ are linear combinations of a selected set of frequencies $\{\lambda_m\}_{m \in M}$ from the original series.
  3. The proof utilizes a convolution with a Fejér kernel $K(u)$ and a Gaussian weight $\Phi(u)$ to isolate specific terms.
  4. By analyzing the ratio of integrals $\frac{\int F(x)|R(x)|^2 \Phi}{\int |R(x)|^2 \Phi}$, the author demonstrates that the "resonance" amplifies the contribution of the selected terms $f(m)$ while suppressing others.

Key Technical Innovation:
The method treats the resonance technique as an additive device. By carefully constructing the resonator using "square-free" frequencies (linear combinations of the building blocks), the author avoids the linear dependency issues that plague Dirichlet's approximation theorem. This allows the selection of a set $M$ with cardinality $M \asymp \log T$ (where $T$ is the interval length), rather than $o(\log T)$.

3. Key Contributions and Main Theorems

Theorem 1.1 (Main Inequality)

The paper establishes a new lower bound for the maximum of the real part of a rotated trigonometric polynomial:
$$\max_{x \in [Y/2, 2T]} \left| \Re(e^{i\beta}F(x)) \right| \ge C \sum_{m \in M} f(m) - \text{Error Terms}$$
Crucially, the error term depends on $M$ and $T$ in a way that allows $M$ to be as large as $O(\log T)$ while keeping the interval length polynomial in $T$. This is a significant improvement over Soundararajan's inequality (1.2), where the error term forced $M$ to be smaller.

Theorem 1.2 (Measure of Large Values)

For polynomials with positive coefficients, the author provides a lower bound not just for the maximum, but for the measure of the set where the function is large. If the sum of selected coefficients is positive, a significant portion of the interval $[Y, T]$ contains points where $\Re(F(x))$ exceeds a certain threshold.

Theorem 1.5 (Complex Coefficients and Kronecker's Theorem)

The paper generalizes the method to trigonometric polynomials with complex coefficients. It connects the resonance method to Kronecker's Approximation Theorem and Chen's quantitative version of it. The result shows that the resonance method is essentially a constructive proof of Kronecker's theorem, providing explicit bounds on the interval length required to achieve a certain magnitude based on the linear independence of frequencies.

4. Results and Improvements

By applying Theorem 1.1 to the Voronoï summation formulas for the divisor and circle problems, the author derives improved $\Omega$-results.

Corollary 1.3 (Improved Bounds):
The new lower bounds for the error terms are:
$$\Delta(x) = \Omega\left( x^{1/4} (\log x)^{1/4} (\log \log x)^{\frac{3}{4}(2^{4/3}-1)} (\log \log \log x)^{-3/8} \right)$$
$$P(x) = \Omega\left( x^{1/4} (\log x)^{1/4} (\log \log x)^{\frac{3}{4}(2^{1/3}-1)} (\log \log \log x)^{-3/8} \right)$$

Comparison with Previous Work:

• Soundararajan (2003): The exponent of the triple logarithm $(\log \log \log x)$ was $-5/8$.
• Sourmelidis (2024): The exponent is improved to $-3/8$.
• Significance: While the improvement in the exponent seems small, the author notes that under current conjectures (supported by probabilistic models from Lamzouri), the power of the triple logarithm is the primary barrier to the "true" maximal order. Reducing the negative power brings the proven lower bound closer to the conjectured true order.

Corollary 3.1 & 3.2:
Similar improvements are obtained for the $k$-divisor problem and the mean square error of the Riemann zeta-function $E(t)$.

5. Significance and Implications

1. Methodological Shift: The paper successfully re-frames the resonance method from a purely multiplicative tool (used for $L$-functions) to a powerful additive tool for trigonometric polynomials. This bridges the gap between analytic number theory techniques used for divisor problems and those used for zeta-function extremes.
2. Optimization of Parameters: By overcoming the limitations of Dirichlet's approximation theorem, the method allows for the selection of a larger set of "resonant" frequencies ($M \asymp \log T$). This directly translates to stronger lower bounds in problems where the error term is governed by the accumulation of many small oscillations.
3. Connection to Kronecker's Theorem: The work highlights that the resonance method is a modern, quantitative realization of Kronecker's approximation theorem. It provides a constructive way to find points where a trigonometric polynomial is large, linking the distribution of frequencies to the length of the interval required.
4. Future Directions: The author suggests that while the current improvement is limited by the power of the triple logarithm, the framework allows for further refinements. Specifically, if one can construct more complex resonators (beyond the "square-free" model used here), even better constants and potentially higher powers of logarithms could be achieved.

In summary, Sourmelidis provides a rigorous technical advancement in the study of extremal values of arithmetic functions, demonstrating that the resonance method, when applied additively, yields superior results to classical approximation theorems in lattice point problems.