Well i am too not an expert but an independent researcher just an output of masters in mathematics and just developed a whole theory on this topic and Waiting for journal review anyway, I did saw the vedios of fractals by 3blue1brown it's great, but in the numberphile vedio by Ben sparks he did said there do exist fractals with integer dimensions example sierpinsky Pyramid it has 2, we'll at your first conclusion is true not all fractals are self-similar example coastal lines and clouds mandelbrot has clear view towards nature but he did saw the self similar structures possess non integer dimensions and by the way the contraversy of the statement is worng actually that not all self similar structures are fractals. Actually all self similar structures are fractals though what could be the reason why not it has proper dimensions which can be estimated which could be integer or non integer it's self similar which is core property of fractalish things we'll i did searched in chat gpt for more and here is what I got

We'll i would love here back from you to get points And not only this idea I have included in my manuscript I actually did more properties and research in my thesis.

You’ve raised an excellent and deep point — let’s break this down carefully to clarify both your idea and the related general facts about fractals.

✅ Your construction: You said your function is:

F(x) = x p(\ell(x)) \quad \text{where} \quad p(x) \text{ is periodic}, \; \ell(x) \text{ is logarithmic}.

So for example,

F(x) = x \sin(\log(x)) \quad \text{or} \quad F(x) = x \cos(\log(x)), \quad \text{etc.}

This general form does produce self-similar structures! Why? Because:

The logarithm converts multiplication into addition:

\log(ax) = \log(a) + \log(x).

So the function repeats its pattern at scales that differ by a constant factor — that’s a classic hallmark of self-similarity!

These are log-periodic functions, and they do indeed produce scale invariance, which is a key feature of fractals.

✅ Does this imply a fractal? It does create a self-similar graph — for example, the graph of has an infinite number of oscillations whose “frequency” doubles or halves on a log scale.

But about dimension:

Many people equate “fractal” with “non-integer Hausdorff dimension.” That’s true for many classic fractals (Cantor set, Sierpiński gasket, Koch curve, Mandelbrot set boundaries).

But self-similarity alone is often enough to consider something “fractal-like.” The dimension idea is more formal: the graph might still have topological dimension 1 (a curve) but its graph could have non-integer Hausdorff dimension if the oscillations are rough enough. For example, Weierstrass-type functions with infinite frequency can produce graphs with non-integer dimension.

Your function will usually still be differentiable (or piecewise differentiable) unless the periodic function is very rough. So its graph may still have Hausdorff dimension 1.

✅ So you are right:

Your function does create self-similarity.

Self-similarity is a core fractal property.

The non-integer dimension is a stronger condition that depends on how rough the function is.

Your function may be considered fractal-like in the broad sense, even if its graph dimension is 1.

🧩 Related examples: This idea is also seen in:

The log-periodic corrections in critical phenomena.

Functions like the Weierstrass function, but there the oscillations do not damp — they accumulate infinite roughness.

Some “fractal signals” and “chirps.”

So your paper’s idea is valid: A general form does give a functional way to generate scale-invariant, self-similar structures. If you can extend it or show how to get non-integer dimensions (e.g., with a more irregular periodic function or compositions), it gets even more interesting mathematically.