# Rotationally Symmetric Lacunary Functions and Products of Centered Polygonal Lacunary Functions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Definitions and Notation

**Theorem**

**1.**

**Definition**

**1.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Definition**

**4.**

## 3. Rotational and Dihedral Symmetry of Centered Polygonal Lacunary Functions

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Constructing Rotationally Symmetric Lacunary Functions

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 5. Renormalized Products of Centered Polygonal Lacunary Functions

**Theorem**

**7.**

**Proof.**

#### 5.1. Renormalized Powers

**Definition**

**5.**

**Theorem**

**8.**

**Proof.**

#### 5.2. Symmetry-Sequence Products

**Definition**

**6.**

**Theorem**

**9.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The example of $f\left(z\right)={\sum}_{n=1}^{\infty}{z}^{n!}$. To render the plot an upper limit, N, on the sum must be chosen. In this case, $N=40$. Panel (

**a**) is the three-dimensional graph of $\left|f\right(z\left)\right|$. Panel (

**b**) shows the corresponding contour plot. Panel (

**c**) is a particularly important representation. In this case, the contour plot is limited to $\left|f\right(z\left)\right|\le 1$. The contour at $\left|f\right(z\left)\right|=1$ is referred to as the unity contour. The part of the unity contour which is a closed curve encircling the origin is called the main unity contour (see Definition 2).

**Figure 2.**The example of lacunary functions generated from the centered polygonal numbers. Panel (

**a**) shows the case for $k=3$, $N=20$. Panel (

**b**) shows the case for $k=5$, $N=20$. Panel (

**c**) shows case for $k=7$, $N=20$. In all cases, the true rotational symmetry and dihedral mirror symmetry are clearly exposed.

**Figure 3.**The example of lacunary functions generated from the Fibonacci sequence. Panel (

**a**) shows the case for $s=3$, $N=20$, and $h\left(n\right)=\mathrm{fib}\left(n\right)$. Panel (

**b**) shows the case for $s=3$, $N=20$, and $h\left(n\right)=\mathrm{fib}(n+1)-1$. Panel (

**c**) shows the same case as the middle panel but now $s=5$.

**Figure 4.**The example of lacunary functions generated from the powers of n. In all cases $s=3$ and $N=20$. Panel (

**a**) shows $h\left(n\right)={(n-1)}^{2}$. Panel (

**b**) shows the case for $h\left(n\right)={(n-1)}^{4}$. Panel (

**c**) shows the case for $h\left(n\right)={(n-1)}^{8}$. Notice the qualitative similarity among these but the qualitative distinctiveness compared with the lacunary functions generated from the Fibonacci sequence. This is because the power sequence “accelerates” faster than the Fibonacci sequence does, even though the rate of increase ultimately favors the Fibonacci sequence.

**Figure 5.**A sequence of related lacunary functions that systematically vary the rate of gap increase. Here, $h\left(n\right)=\u230a(n-1)+\frac{{(n-1)}^{2}}{w}\u230b$. In all cases, $k=3$ and $N=200$. Across the top row from left to right: (

**a**) $w=1$, (

**b**) $w=3$, and (

**c**) $w=6$. Across the bottom row from left to right: (

**d**) $w=8$, (

**e**) $w=9$, and (

**f**) $w=500$. At low values of w (rapid growth of the gap), one sees significant qualitative changes in the graphs for modest changes in w. At higher values of w, the qualitative features of the graph stabilize. There is a general trend from higher convexity of the domain bounded by the the main unity contour to lower convexity. (Note: a very small positive value ($1\times {10}^{-6}$) is added to w during the rendering to ensure $h\left(1\right)=1$ when $w=1$).

**Figure 6.**Renormalized (

**a**) and non-renormalized (

**b**) cubic power of ${f}_{10}^{\left(k\right)}$ for $k=3$, i.e., ${}^{3}\phantom{\rule{-0.166667em}{0ex}}{R}_{10}^{\left(3\right)}\left(z\right)$ and ${\left({f}_{10}^{\left(k\right)}\right)}^{3}$, respectively. In general, renormalization produces a more convex unity level set as compared to the non-renormalized counterpart.

**Figure 7.**Renormalized powers (first, $j=1$, to sixth, $j=6$) of ${}^{j}\phantom{\rule{-0.166667em}{0ex}}{R}_{10}^{\left(3\right)}\left(z\right)$ ((

**a**) through (

**f**) respectively). Three-fold symmetry is maintained the function narrows in towards the symmetry axes as the power increase, i.e., the convexity of the main unity contour is increased.

**Figure 8.**The product of the sequence of functions ${}^{23}\phantom{\rule{-0.166667em}{0ex}}{K}_{10}^{\left(3\right)}\left(z\right)$. The three-fold rotational symmetry is maintained in this case, although the structure of the unity level set becomes more elaborate.

**Figure 9.**The product of the sequence of functions ${}^{M}\phantom{\rule{-0.166667em}{0ex}}{K}_{10}^{\left(1\right)}\left(z\right)$ for $M=1$ through $M=6$ ((

**a**) through (

**f**) respectively).

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Mork, L.K.; Sullivan, K.; Vogt, T.; Ulness, D.J.
Rotationally Symmetric Lacunary Functions and Products of Centered Polygonal Lacunary Functions. *Fractal Fract.* **2020**, *4*, 24.
https://doi.org/10.3390/fractalfract4020024

**AMA Style**

Mork LK, Sullivan K, Vogt T, Ulness DJ.
Rotationally Symmetric Lacunary Functions and Products of Centered Polygonal Lacunary Functions. *Fractal and Fractional*. 2020; 4(2):24.
https://doi.org/10.3390/fractalfract4020024

**Chicago/Turabian Style**

Mork, L. K., Keith Sullivan, Trenton Vogt, and Darin J. Ulness.
2020. "Rotationally Symmetric Lacunary Functions and Products of Centered Polygonal Lacunary Functions" *Fractal and Fractional* 4, no. 2: 24.
https://doi.org/10.3390/fractalfract4020024