I asked Gemini the following:
How does the self convolution of a set A differ from the Cartesian product of the set with itself (A x A)?
This was the response:
To understand the difference between the Cartesian product of a set and its self-convolution, we have to look at what mathematical object each operation produces. In short: the Cartesian product generates a new set of raw pairs, while the convolution generates a sequence that counts the frequencies of their sums.
Here is a full breakdown of how these two concepts differ and how they ultimately connect.
1. The Cartesian Product: Structural Pairing
The Cartesian product of a set $A$ with itself, denoted as $A \times A$, is an operation strictly within set theory.
It takes the elements of $A$ and creates a new set containing every possible ordered pair $(x, y)$, where both $x$ and $y$ are elements of $A$.
$$A \times A = \{(x, y) \mid x \in A \text{ and } y \in A\}$$
- Nature: It is purely structural. It does not add, multiply, or combine the values of the elements; it merely groups them side-by-side in a 2D grid.
- Size: If set $A$ contains $k$ elements, the Cartesian product $A \times A$ will always contain exactly $k^2$ elements.
2. The Self-Convolution: Frequency of Sums
Strictly speaking, you cannot convolve a "set" directly; convolution is an operation applied to functions or sequences. When mathematicians refer to the convolution of a set $A$, they are actually referring to the discrete convolution of the set's indicator sequence (also called the characteristic function).
Let $1_A(x)$ be a sequence where the value is $1$ if $x$ is in set $A$, and $0$ if it is not. The self-convolution of this sequence produces a new sequence, $c_n$:
$$c_n = \sum_{i+j=n} 1_A(i) \cdot 1_A(j)$$
- Nature: It is an arithmetic and combinatorial operation. The resulting sequence $c_n$ tells you exactly how many times the integer $n$ can be expressed as the sum of two elements from set $A$.
- Size: The result is not a finite set of pairs, but an infinite sequence of integers (though it will be mostly zeros if $A$ is a finite set).
3. The Core Difference
The Cartesian product asks: "What are all the possible ways to pick two elements from this set?"
The Convolution asks: "If I add those two elements together, how many pairs produce a sum of $n$?"
If you imagine the Cartesian product as a square grid of pairs, the convolution is the result of drawing diagonal lines through that grid (where $x + y = n$) and counting how many pairs fall on each diagonal.
4. A Concrete Example
Let’s define a set of integers: $A = \{1, 2, 4\}$
The Cartesian Product ($A \times A$) creates a set of $3^2 = 9$ ordered pairs:
$A \times A = \{(1,1), (1,2), (1,4), (2,1), (2,2), (2,4), (4,1), (4,2), (4,4)\}$
The Self-Convolution maps the sums of these pairs to their frequencies. Let's add the elements of each pair together:
- Sums to 2: $(1,1) \rightarrow$ 1 pair
- Sums to 3: $(1,2), (2,1) \rightarrow$ 2 pairs
- Sums to 4: $(2,2) \rightarrow$ 1 pair
- Sums to 5: $(1,4), (4,1) \rightarrow$ 2 pairs
- Sums to 6: $(2,4), (4,2) \rightarrow$ 2 pairs
- Sums to 7: None $\rightarrow$ 0 pairs
- Sums to 8: $(4,4) \rightarrow$ 1 pair
The resulting convolution sequence $c_n$ (starting from index $n=0$) would look like this:
$0, 0, 1, 2, 1, 2, 2, 0, 1, 0, 0, 0...$
5. The Generating Function Connection
Because you are working with tools like SageMath, the most elegant way to see the relationship between these two concepts is through generating functions.
If you turn set $A$ into a polynomial where the elements of the set are the exponents, you get:
$$f(x) = x^1 + x^2 + x^4$$
When you square this polynomial (which is the algebraic equivalent of the Cartesian product), you get:
$$f(x)^2 = (x^1 + x^2 + x^4) \cdot (x^1 + x^2 + x^4)$$
$$f(x)^2 = x^2 + 2x^3 + x^4 + 2x^5 + 2x^6 + x^8$$
Look at the coefficients of the expanded polynomial: $1, 2, 1, 2, 2, 1$. These are the exact non-zero values of our convolution sequence. The Cartesian product dictates the terms being multiplied, but the mathematical simplification of gathering like-terms is the act of convolution.
