Chowla Cosine Problem: Extremal Cosine Sums

• Chowla Cosine Problem is defined by examining extremal negative values of finite cosine sums over distinct positive integers.
• Recent breakthroughs have established the first uniform polynomial lower bound (with exponent at least 1/12), advancing earlier logarithmic bounds.
• Analytical techniques include Fourier analysis, combinatorial methods, and convolution strategies, linking the problem to conjectures in number theory and transcendence.

The Chowla Cosine Problem comprises a family of extremal questions at the interface of harmonic analysis, number theory, and arithmetic combinatorics. It centers on understanding the cancellation phenomena, extremal values, and sign correlations of finite trigonometric sums of the form $\sum_{j=1}^n \cos(a_j\theta)$ as the frequencies $a_j$ run through sets of positive integers, and its numerous variants have deep connections to problems in analytic number theory (especially the Chowla and Sarnak conjectures), Diophantine approximation, and transcendence theory.

1. Classical Chowla Cosine Problem: Extremal Minimum of Cosine Sums

Given $n\in\mathbb{N}$ and distinct positive integers $a_1<\cdots<a_n$, define the length-$n$ cosine sum

$f(\theta)=\cos(a_1\theta)+\cdots+\cos(a_n\theta).$

Chowla posed the problem of determining, or bounding, the largest possible minimum of such a sum as the $a_j$ vary: $$\lambda(n):=-\sup_{a_1<\cdots<a_n} \min_{\theta\in\mathbb{R}} f(\theta).$$ This “largest minimum” (with a sign convention so $\lambda(n)>0$) quantifies how negative a sum of $n$ cosines, at arbitrary frequencies, is forced to become.

Exact values for small $a_j$0 are known:

• For $a_j$1, $a_j$2, realized uniquely (up to scaling) by $a_j$3: $a_j$4.
• For $a_j$5, $a_j$6, with the extremizer given by $a_j$7: $a_j$8 (Mercer, 2012, Mercer, 2017).

The main analytic approach uses trigonometric and arithmetic structure, sampling over arithmetic progressions on the circle, and leveraging orthogonality and averaging principles. For instance, sampling $a_j$9 over appropriate roots of unity allows one to force certain cosine terms to fixed values, reducing the problem to a finite-dimensional analysis even as $n\in\mathbb{N}$0 range over infinitely many possibilities (Mercer, 2017).

For general $n\in\mathbb{N}$1, Chowla conjectured that

$n\in\mathbb{N}$2

as $n\in\mathbb{N}$3. However, the best general lower bound for $n\in\mathbb{N}$4 terms has seen incremental improvement: Roth, McGehee–Pigno–Smith, and Konyagin established logarithmic-type growth, Bourgain raised this to quasi-polynomial, and recent work provides the first polynomial lower bound (Bedert, 5 Sep 2025).

2. Lower Bounds, Polynomial Exponents, and Recent Progress

The contemporary refinement of the Chowla Cosine Problem is to find sharp lower bounds for

$n\in\mathbb{N}$5

over all $n\in\mathbb{N}$6-element sets $n\in\mathbb{N}$7.

The breakthrough result (Bedert, 5 Sep 2025) is the following: For all $n\in\mathbb{N}$8,

$n\in\mathbb{N}$9

for some absolute constant $a_1<\cdots<a_n$0, i.e.

$a_1<\cdots<a_n$1

for all $a_1<\cdots<a_n$2 of size $a_1<\cdots<a_n$3. This is the first uniform polynomial lower bound for the Chowla Cosine Problem, and it improves significantly on all earlier bounds (logarithmic, quasi-polynomial, or powers of $a_1<\cdots<a_n$4). The proof synthesizes additive combinatorics (particularly structure and energy of sets with small negative minimum), convolution methods on groups, Fourier-analytic positivity, and iterative phase-amplification mechanisms.

The argument proceeds by:

• Translating the problem to positivity of a real mean-zero trigonometric polynomial;
• Using Roth-type combinatorial lemmas to produce large subsets with structured additive overlap;
• Amplifying non-symmetric components through convolution and complex phase rotation;
• Extracting a quantitative relation between the lower bound and the set size.

This leaves a significant gap relative to Chowla's conjectured square-root bound, but is a major step forward.

3. Extremal Sets, Finite Computations, and Explicit Extremizers

For small $a_1<\cdots<a_n$5, extremal sets of exponents can be determined explicitly. The methods systematically reduce the infinite search over all possible $a_1<\cdots<a_n$6-element sets to a finite (but intricate) analysis, by normalization (gcd reductions), symmetry (reflection and translation), and arithmetic case distinction. For example, for $a_1<\cdots<a_n$7, (Mercer, 2012, Mercer, 2017) divide all possible triples into arithmetic types, deal separately with resonant configurations (e.g., $a_1<\cdots<a_n$8, $a_1<\cdots<a_n$9), and resolve the rest by weighted average principles and second-moment arguments.

These approaches combine

• explicit trigonometric minimization,
• equispaced circle sampling,
• orthogonality arguments,
• and, for moderate $n$0, direct computation.

By these means, the unique (up to symmetry) extremal sets for $n$1 and $n$2 are established. For $n$3 and beyond, structural decompositions and heuristic roadmaps exist, but explicit computation has proved much harder.

4. Cosine Sign Correlation: Sign Patterns and Probability

A key probabilistic variant is the cosine sign correlation problem: for fixed $n$4, let $n$5 be uniform on $n$6, and consider the probability that all $n$7 share the same sign: $n$8 The minimum of $n$9, as the $f(\theta)=\cos(a_1\theta)+\cdots+\cos(a_n\theta).$0 vary, is explicitly computed for $f(\theta)=\cos(a_1\theta)+\cdots+\cos(a_n\theta).$1 and $f(\theta)=\cos(a_1\theta)+\cdots+\cos(a_n\theta).$2:

• $f(\theta)=\cos(a_1\theta)+\cdots+\cos(a_n\theta).$3: The minimum is $f(\theta)=\cos(a_1\theta)+\cdots+\cos(a_n\theta).$4, uniquely realized (up to scaling) by $f(\theta)=\cos(a_1\theta)+\cdots+\cos(a_n\theta).$5
• $f(\theta)=\cos(a_1\theta)+\cdots+\cos(a_n\theta).$6: The minimum is $f(\theta)=\cos(a_1\theta)+\cdots+\cos(a_n\theta).$7, uniquely by $f(\theta)=\cos(a_1\theta)+\cdots+\cos(a_n\theta).$8 for both, see (Dou et al., 2022).

Unexpectedly, the pattern does not naively extend beyond $f(\theta)=\cos(a_1\theta)+\cdots+\cos(a_n\theta).$9; for $a_j$0 the set $a_j$1 yields a smaller correlation than $a_j$2, as verified by computation. This phenomenon demonstrates that the structure of extremal cosine sign correlations is more subtle than hierarchical geometric progression.

These results use a blend of frequency arithmetic, finite sampling, Fourier methods, and symmetry reduction, and connect to questions in spectral theory and combinatorial problems like the lonely runner.

5. Chowla Cosine Problem and Connections to Number Theory and Transcendence

The significance of the Chowla Cosine Problem is magnified by its deep links to

• the Chowla and Sarnak conjectures on shift correlations of the Möbius and Liouville functions;
• product formulas and special values in transcendence theory, such as the linear independence measures of Chowla–Selberg periods (Zudilin, 25 Aug 2025);
• explicit class number/congruence relations involving real quadratic fields, units, and Bernoulli or Euler numbers (e.g. the Ankeny–Artin–Chowla conjecture, AAC-type congruences, and p-adic $a_j$3-functions (Fellini, 2024, Fellini et al., 2023));
• the arithmetic of complex multiplication and the Chowla–Selberg formula (both modulus and argument), as seen in explicit trigonometric phase phenomena in the CM values of the Dedekind eta function (Cohen, 1 Jun 2026).

While explicit trigonometric product or sum identities (cosine products, argument formulas) are typically not the direct focus of modern analytic work on the extremal minimum problem, the underlying arithmetic structure remains the same: cancellation, periodicity, and “algebraic randomness” versus subtle arithmetical rigidity.

6. Fourier-Analytic and Ergodic Variants

A Fourier-theoretic variant—sometimes also called the “Chowla cosine problem”—concerns shifted autocorrelations of periodic or arithmetic functions (often Möbius or Liouville), with focus on cancellation of Fourier coefficients or their products. For example, the “periodic Chowla conjecture” for Möbius, proved in a model setting by discrete Fourier transform methods, asserts: $a_j$4 with $a_j$5 arbitrary, where $a_j$6 is the Möbius function extended modulo a large prime $a_j$7 (Carella, 2022). The main engine is the convolution theorem for Fourier transforms on the finite group, with exponential sum bounds at its heart.

Similarly, ergodic-theoretic and logarithmically averaged forms of the Chowla conjecture (and its extensions to more general multiplicative functions) are crucial for the analytic and probabilistic understanding of randomness and structure in arithmetic (Tao, 2015, Tao et al., 2017).

7. Broader Arithmetic and Transcendence Context

A family of results rooted in the Chowla–Selberg formula, its argument refinement, and generalizations to products and congruences for quadratic fields and modular functions (e.g., (Cohen, 1 Jun 2026, Zudilin, 25 Aug 2025, Medjedovic, 2020, Fellini, 2024, Fellini et al., 2023, Reinhart, 2024)) reveals the persistence of trigonometric, cyclotomic, and periodic structure—sometimes as explicit cosine or root-of-unity factors—in the arithmetic of special $a_j$8-values, class numbers, and periods.

Many of these explicit “cosine”-type components are now understood as reflections of deeper class field theory and transcendence-theoretic properties, but their appearance continues to motivate and inspire both theoretical analysis and computational exploration.

References (arXiv IDs)

Topic Area (Summary)	Principal Papers
Exact values for small $a_j$9 and methodologies	(Mercer, 2012, Mercer, 2017)
Polynomial lower bounds	(Bedert, 5 Sep 2025)
Cosine sign correlation (probabilistic model)	(Dou et al., 2022)
Fourier/ergodic periodic Chowla variant	(Carella, 2022)
Relations to p-adic $$\lambda(n):=-\sup_{a_1<\cdots<a_n} \min_{\theta\in\mathbb{R}} f(\theta).$$0-functions, AAC	(Fellini, 2024, Fellini et al., 2023, Reinhart, 2024)
Chowla–Selberg periods, transcendence	(Zudilin, 25 Aug 2025, Cohen, 1 Jun 2026, Medjedovic, 2020)

This spectrum of research demonstrates the deep interplay between harmonic analysis, combinatorial and additive number theory, automorphic functions, and transcendence theory that animates the modern landscape of the Chowla Cosine Problem and its variants.