^{1, 2}

^{1}

^{2}

We abstract the definition of the Costas property in the context of a group and study specifically dense
Costas sets (named Costas clouds) in groups with the topological property that they are dense in themselves:
as a result, we prove the existence of nowhere continuous dense bijections that satisfy the Costas property on

Costas arrays are square arrangements of dots and blanks such that there is exactly one dot per row and column (i.e., permutation arrays), and such that no four dots form a parallelogram and no three dots on the same straight line are equidistant. They arose in the 1960s in connection with the development of SONAR/RADAR frequency-hopped waveforms with ideal autocorrelation properties [

Golomb rulers are 1D analogs of Costas arrays: they are linear arrangements of dots and blanks such that no distance between pairs of dots is repeated. The term “ruler’’ arises from the equivalent visualization as a ruler with markings at the integers, where the dots correspond to those markings that get selected. Golomb rulers are, in fact, older than Costas arrays themselves, originally described as Sidon sets, namely, sets of integers whose pairwise sums are all distinct (a moment's reflection shows that the two definitions are completely equivalent).

A deeper study of the Costas property, which we attempt in this work, reveals that it requires surprisingly little, namely, only an underlying algebraic group structure. After recognizing that some parts of the definition of a Costas array are actually additional, peripheral requirements, imposed for convenience by the nature of the engineering application, but not essential (even for the application itself), we show that Costas arrays and Golomb rulers are essentially instantiations of the same concept/property over different groups.

Though this approach opens the door for the study of “exotic’’ Costas structures on arbitrary algebraic groups (possibly nonAbelian), we restrict our attention almost immediately on groups with the analytic property of being dense in themselves, and in particular the fields

An example of a Costas array of order 27: note the haphazard positioning of its dots.

In a previous work [

In this work, which can be construed as a continuation of [

We denote the set of the first

In this section we provide an overview of the Costas property and, at the same time, an appropriate generalization to an arbitrary group; usually the Costas property is defined on

Let

Let

The elements of

Let

The usual definition of a Costas bijection uses

Similarly, the definition of a Golomb ruler uses

Historically, the Costas property has been construed to include bijectivity: this is because the original engineering application that introduced Costas arrays does not benefit any further from nonbijective Costas sets, namely, Costas arrays with two or more dots on the same row or column, though such Costas arrays can still be used successfully [

A direct consequence of the definition is that for all

We finally specialize the definition of a Costas set on groups that are dense in themselves.

Let

In particular, Costas clouds can exist in

Two algebraic construction methods exist for Costas permutations [

Let

The reason for the presence of

Let

Golomb rulers, though bearing the name of Professor S. Golomb, were originally described by W. C. Babcock in the context of an application in telecommunications [

Definition

Let

The sum version corresponds to Sidon sets, the difference version to Golomb rulers: they are obviously equivalent. What is the relation between

Though they will not be needed further in this work, and for the sake of completeness only, we also present some construction methods for Golomb rulers. Note that, contrary to the case of Costas arrays, where the order specifies the number of dots, the definition of a Golomb ruler does not relate the length to the number of markings in any way. Needless to say, construction methods for Golomb rulers considered to be of interest tend to produce reasonably densely populated Golomb rulers, and, in particular, families that asymptotically satisfy (

For every prime

The approximate asymptotic length of such a Golomb ruler with

Let

These rulers are of length (at most)

Let

In addition, the set of

These rulers are optimal: they have

The previous constructions work only when the number of markings is a (power of

For any

This ruler has

The following two examples use the indicator function of a dense subset

Let

We will denote the set of irrational numbers by

We now show that

Since

We finally need to show that

The previous idea can be successfully generalized to yield an expanded family of nowhere continuous functions with the Costas property if

Let

The set

The equation

The equation

Since

Finally, we need to show that

If

If

Although both constructions above are nowhere continuous, their behavior is not as “wild’’ as one might have hoped for: their graph is entirely included within two smooth curves, given by the equations

Our foremost intention here is to prove that bijective Costas clouds exist; to carry out the proof, we will need some background.

Let

A detailed study of this equation can be found in [

The solution

For all

(i) Setting

(ii) Setting

(iii) Setting

(iv) Setting

Expressing the rational

Assume now that

Assume that

Assume now that

Cauchy's functional equation, despite its simplicity, has been playing a prominent role in analysis: Hilbert's 5th problem essentially proposes a generalization of this equation, while an important area of study is the Hyers-Rassias-Ulam stability of this equation (or slight variants thereof) [

There exist solutions of Cauchy's equation that are nowhere continuous bijections/injections, everywhere dense on the real plane.

Consider

The theorem does not rely on the exact nature of

Bijective Costas clouds on

Consider a nowhere continuous bijection

Costas clouds on

It has been suggested that the success of the Welch method (Theorem

Let

Since the Welch method can be successfully generalized on the real line, can the same be done for the Golomb method (Theorem

Let

Based on Theorem

Bijective Costas clouds on

Consider the function

Costas clouds on

But we can also apply another simple transformation, again based on Theorem

Consider the function

An algorithm to construct Costas bijections on the set

The construction proposed in Theorem

Enumerate

Draw the horizontal line through

Draw the horizontal lines through

The construction is possible because, for every stage and every square, we are called to choose a rational point in this square so that finitely many constraints are satisfied: this is always possible as there are infinitely many rational points in a square. The resulting function clearly satisfies the Costas property, it is injective between its domain and its range, and its graph is everywhere dense in

The construction above can be modified to yield bijective rational Costas clouds on the entire

The square grid described in the theorem has the property that the squares in a given stage are all of the same size, and that any two squares either share a common boundary or else one contains the other. Neither of these properties is necessary, strictly speaking: for example, we could have used a grid of the form

The proof does not depend on the exact nature of

Infinite Golomb rulers have been studied in the past [

Let

The idea of the proof is essentially the same as in Theorem

To ensure that the ruler is dense in

Any countably infinite set, such as

Let

Consider

Note that, for any

Having investigated the Costas property on

Cauchy's functional equation (

The proofs of the extended Welch (Theorem

We need to study the properties of a function

Applying the real argument twice (on the real and the imaginary numbers), we obtain

We now need to ensure that

Assuming

It remains to be seen that such a nonlinear function can be constructed. To achieve this, we proceed as in the proof of Theorem

The extensions of both the Golomb and the Welch construction in

Let

The proof is a verbatim repetition of the proof of Theorem

Let

The proof is a verbatim repetition of the proof of Theorem

The Costas property is very “unalgebraic’’, as most of the usual algebraic operations fail to preserve it: for example, the composition of two Costas functions is, in general, not a Costas function, and the same goes for their sum, product, and so forth. We now take a look at two operations that do preserve the Costas property.

Group isomorphisms can be composed with Costas sets to yield new Costas sets, or to divide Costas sets into equivalence classes. This seemingly simple result has some important consequences; in particular, it can be used to prove the existence of new Costas clouds.

Let

Let

Bijective Costas clouds on

We use Theorem

It all then comes down to finding the appropriate function

It follows that

Let

The proof is practically obvious: letting

For example, let

As another example, let now

As a final caveat, however, note that Theorem

We abstracted the Costas property and stated it in the context of an arbitrary (possibly even nonAbelian) group. As a consequence, Costas arrays and Golomb rulers were both found to be Costas sets, instantiations of the Costas property over different groups. The bijectivity condition in the case of Costas arrays is an additional, peripheral requirement, not directly related to the Costas property. We did not further pursue the direction of the study of Costas sets over nonAbelian groups, which we leave as future work, but turned our attention to groups with the analytic property of being dense in themselves instead (such as

We first constructed explicit examples of real, nowhere continuous bijections whose graphs satisfy a constrained form of the Costas property (over rational or algebraic displacements only, i.e.), using the indicator function of the rationals or of the algebraic numbers as a building block. Furthermore, we constructed real Costas clouds, based on Cauchy's functional equation: these are perhaps what first springs into one's mind when considering possible generalizations of Costas arrays in the continuum, due to the very haphazard positioning of their dots. These Costas clouds led to the generalization in the continuum of the two main generation methods for Costas permutations, namely, the Welch and the Golomb construction. These functions are highly nontrivial to construct, and their existence nontrivial to prove. We also considered rational Costas clouds, which were easier to construct thanks to the enumerability of the rationals.

Similarly, we proved the existence of (countably and uncountably) infinite Golomb rulers in a (finite of infinite) interval of the real line, that can optionally be constructed so that they have the extra property of being everywhere dense in this interval. We also noted that, though “infinite dense’’ Golomb rulers have appeared in literature before, the word “dense’’ had an entirely different meaning and was not used in the analytic sense.

Both uncountable constructions (Costas clouds and Golomb rulers) relied on two ideas, namely, the consideration of

The extension to complex numbers of the construction methods for Costas clouds we presented was possible and necessitated a slight modification of the Cauchy functions used. The main obstacle, however, was the multivalued nature of the exponential function that is involved in both the Golomb and the Welch method: in order to re-establish bijectivity, the Costas sets produced by these methods needed to be appropriately redefined over equivalence classes of points (or distinct representatives thereof).

Despite all of our efforts, the real Costas clouds we were able to construct through the extensions of the Golomb and the Welch method were limited on genuine subsets of the real plane (the upper half plane, an infinite strip, etc.). A bijective Costas cloud on the entire real plane was finally constructed by a new general construction principle applicable on any group, namely, that the composition of a Costas set with a group isomorphism results in a new Costas set: in this particular instance, the graph of a suitably constructed smooth Costas function on

The ideas presented here have potentially far reaching consequences, which we intend to investigate as future work: for example, Costas sets over arbitrary (even noncommutative) groups have never been studied (to the best of our knowledge); how would Costas arrays over GL

The author is indebted to Professor Nigel Boston (School of Mathematics, University College Dublin) for his helpful suggestions regarding Theorem