Iterated Remainder Sets

• Iterated remainder sets are recursively defined collections of remainders produced by sequentially applying the modulo operation, foundational for understanding Pierce expansions and residue dynamics.
• They provide quantitative bounds and asymptotic growth measures, with explicit factorial lower bounds and linear upper bounds that describe the decay in set sizes over iterations.
• Their analysis uncovers nonmonotonic behavior in remainder functions and connects classical number theory with modern combinatorial and dynamical interpretations.

Iterated remainder sets are algebraic or combinatorial constructs arising from the systematic application of the remainder (modulo) operation in a recursive or iterative fashion, often informed by problems in number theory, combinatorics, dynamical systems, and valuation theory. Their paper encompasses the structure, asymptotic growth, and dynamical behaviors that manifest as one applies the remainder process successively to integers, sets, or residue classes. Iterated remainder sets appear naturally in the analysis of Pierce expansions, the recursive decomposition of number systems, the paper of orbits in dynamical systems, and the enumeration of mappings with specific cycle constraints.

1. Recursive Construction and Pierce Expansions

The foundational construction begins with the set

$$S(n) = \{ n \bmod k : 1 \leq k \leq \lfloor n/2 \rfloor \},$$

counting the distinct remainders (s(n)) produced by dividing n by all integers up to n/2. The process generalizes recursively: $$S_0(n) = \{1, 2, \dots, \lfloor n/2 \rfloor \}; \qquad S_{j+1}(n) = \{ n \bmod k : k \in S_j(n) \setminus \{0\} \},$$ with $s_j(n) = |S_j(n)|$. This recursive operation defines iterated remainder sets (S_j(n)), each corresponding to the next level of applying the remainder process.

This schema is deeply connected to Pierce expansions, specifically the iteration

$a_0 = a, \quad a_{j+1} = n \bmod a_j,$

with the goal of determining the minimal $t$ (denoted $P(n, a)$) such that $a_t = 0$. The iterated remainder sets serve as a combinatorial representation of all possible terminal values achievable via these recursive chains; notably, the degeneration of $S_{t+1}(n)$ to $\{0\}$ precisely marks the maximum length of a Pierce expansion chain for $n$ $[2508.20853]$.

2. Quantitative Bounds and Asymptotic Growth

The paper establishes explicit bounds for the normalized size of these iterated remainder sets: $$\frac{1}{(j+2)!} \leq \liminf_{n \to \infty} \frac{s_j(n)}{n} \leq \limsup_{n \to \infty} \frac{s_j(n)}{n} \leq \frac{1}{j+2}$$ for all $j \geq 0$ $[2508.20853]$. Therefore, as the number of iterations increases, the relative size of the iterated remainder set decays rapidly—a factorial lower bound, but linear upper bound in $n$.

While $s_1(n) = s(n)$ satisfies $s(n) = c \, n + O(n/(\log n \log\log n))$ with an explicit constant $c \approx 0.2296$, higher iterates exhibit sparser growth subject to the above inequalities. It is further demonstrated (through combinatorial and analytic tools) that, for $j \geq 2$, the distribution between $\liminf$ and $\limsup$ can be nontrivial; numerical experiments suggest potential sensitivity and possible failure of convergence for the normalized ratios.

3. Analysis of Incremental Changes and Nonmonotonicity

A significant discovery is the erratic behavior of $s(n)$ under increment—while $s(n+1) \leq s(n) + 1$, the function may drop by as much as $O(\log\log n)$, with

$$\liminf_{n \to \infty} \big( s(n+1) - s(n) \big ) = -\infty.$$

This means the function, although linearly trending, experiences abrupt gaps and nonmonotonic changes $[2508.20853]$. This phenomenon arises from the discontinuity of divisor structure as $n$ advances, with many remainders "disappearing" due to the loss of divisibility by certain $k$.

The set of "not transferred elements" $T(n, n+1)$, rigorously characterized in the paper, structurally explains these jumps, encapsulating divisors $k$ for which $n \bmod k$ is not preserved under the increment $n \to n+1$.

4. Connections to Classical and Modern Problems

Iterated remainder sets intersect classical themes in arithmetic and combinatorics:

• Pierce expansions: By encoding possible chain lengths and terminal values, iterated remainder sets provide a means to quantify and analyze the "n mod a" process structurally. The paper demonstrates (Lemma 5.1) that every $r \in S_j(n)$ can be realized as the $j+1$th remainder in a specific Pierce expansion chain.
• Dynamical processes: The recursive construction aligns with dynamics observed in continued fraction expansions, Euclidean algorithms, and iterative digit systems, where the remainder process captures self-similarity or fractal-like behavior.
• Enumeration of restricted mappings: The paper of A-mappings uses remainder terms in generating function coefficients to account for architectures where cyclic decomposition is restricted, trending toward power-law reduction in asymptotics $[2311.14952]$.

5. Tabular Comparison of Iterative Structures

Iteration Stage	Cardinality Bound	Process Representation
$S_1(n)$	$c \cdot n + O(n/(\log n\log\log n))$	$n \bmod k,\, k\leq\lfloor n/2\rfloor$
$S_j(n)$	$(j+2)^{-1} n$ (upper bound)	$n \bmod k$, $k\in S_{j-1}(n)\setminus\{0\}$
Pierce Chain	$P(n)$ iterations to 0	$a_{j+1} = n \bmod a_j$

This table illustrates the recursive progression, complexity scaling, and the underlying process induced by the remainder operator.

6. Research Directions and Open Questions

The bounds for the size of iterated remainder sets raise several unresolved issues:

• Limiting behavior: The gap between $\liminf$ and $\limsup$ for normalized sizes leaves open the existence of an actual limit for $s_j(n)/n$ as $n\to\infty$ when $j\geq 2$.
• Structural characterization: The mechanism for abrupt drops in $s(n)$, while understood combinatorially via non-transferred remainders, could be further explored in terms of divisor patterns and random models.
• Combinatorial dynamics: The recurrence relations and connections to expansions (such as Pierce or continued fraction chains) suggest underlying algebraic or dynamical system behaviors, potentially linking to more general structures in dynamical number theory or symbolic dynamics.

7. Significance and Applications

The concept of iterated remainder sets offers an analytic framework for capturing the complexity inherent in recursive division processes. Their quantitative analysis is crucial for understanding algorithms based on remainder operations (such as Euclidean descent, modular expansions, mapping enumeration with constraints), the distribution of residues, and the underlying arithmetic or combinatorial structure. The established bounds and observed phenomena elucidate both local irregularities and global trends, with implications for computational number theory, dynamical systems, and the analysis of algorithms rooted in divisibility and modular operations.

In summary, iterated remainder sets formalize the recursive remainder operation and serve as a bridge connecting classical number-theoretic algorithms, combinatorial enumeration strategies, and modern investigations into the behavior of arithmetic and dynamical recursive processes $[2508.20853]$.