Cornu Spirals and the Triangular Lacunary Trigonometric System

Round 1

Reviewer 1 Report

I enjoyed reading the study presented in your paper. The pattern arising from triangular numbers and their periodic patterns enforced by the parameter n are interesting.

I recommend some improvement in the presentation of the paper, to help the reader better follow the analysis. Here are some comments/questions:

Page 1

Line 18: I would not refer to (1) as an “equation”, it is simply a definition.

Lines 19-20: does this sentence still references to paper [1]? If so, I would specify that r=2.

Line 21: I know that you provide a reference for triangular numbers, but maybe it is better to include their definition in the text of your paper.

Page 2

Line 25: I would specify the variable w.r.t. which the partial sums are 4n-periodic. Is it q or is it N?

Formula 3: the notation is not consistent with Formula 2. Why the parenthesis (e^{i\pi}) ?

Line 46: f_{1,1} (without the superscript N) is not defined until page 7, and obviously is not a proper “summation” as the infinite series does not converge in the usual sense (unless Cesaro summation is used, but in that case it should be explicitly stated). So better to refer to its definition and refer to it as a divergent series

Line 52: the expression “dense array” is not standard and slightly misleading.  I would say something along the lines of “ ... whose singularities accumulate to the boundary of their domain of definition”.

Page 3

Line 85: Along with Berry-Goldberg I would also cite the work of Hardy-Littlewood [7], Mordell (The approximate functional formula for the theta function, 1923), Fedotov-Klopp (An exact renormalization formula for Gaussian exponential sums and applications), and Cellarosi-Marklof (Quadratic Weyl sums, automorphic functions, and invariance principles).

Page 4

Line 93: what does “generic” mean?

Line 98: Are you using a chi-squared test? For the non-experts (like me) could you add a few words or a reference?

Page 7

Lines 130-131: Can you say something about the convergence of these series? Jacobi’s theta function \sum_{n=-\infty}^{\infty} e^{\pi i (n^2z+ n w)} converges when z is complex and has strictly positive imaginary part. In your case z=q/2n is a real number. The argument of “completing the square” can of course be done for finite sums.

Author Response

 Please see the attachment. Our response to both reviewers is included in the attached file.

Author Response File: Author Response.pdf

Reviewer 2 Report

This paper addresses the original work by Coutsias et al. on the lacunary of trigonometric systems and their relation to the Fresnel integrals and the Cornu spirals.

The paper is interesting for a general audience, seems technically correct and is written in an assertive and  professional style.

The review of the literature covers relevant contributions, but the reviewer recommends having a more extended list of works. 

Recently some connection with fractional calculus and the use of Mathematica was proposed and, therefore, it would interesting to include.

In some figures, the use of colour to distinguish between different traces would benefit the visualization.

Author Response

 Please see the attachment. Our response to both reviewers is included in the attached file.

Author Response File: Author Response.pdf