Apple Bread Challenge Problem 2

library(tidyverse)

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library(viridis)

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library(gganimate)

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library(gifski)

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As at each time ‘n’, the the set \(A_n\) gets \(K\) new elements added to it and \(K-1\) elements subtracted from it, so the size of the set increases by 1 at each step.

A_n = function(A = c(0), K = 3){
  B = c()
  for (m in 1:K) {
      l = 1
      for (j in A){
        B[l] = j-1
        B[l+1] = j+1
        l = l + 2
      }
      C = setdiff(B,A)
      V = sample(C,1,replace = T)
      A[length(A) + 1] = V
    }
    for (n in 1:(K-1)){
      R = sample(A,1,replace = T)
      A = setdiff(A,R)
    }
    return(A)
  }

We plot the Data for n = 100, as n = 1000 produces \(A_n\) that lokks almost like a straight line in the plot.

A = c(0)
K = 3
Data_An = data.frame(Trial = c(0), A_n = c(0))
for (i in 1:100) {
  A = A_n(A, K)
  for (p in 1:length(A)){
    Data_An = Data_An %>% add_row(Trial=i, A_n = A[p])
  }
}

ggplot(Data_An) + geom_point(aes(Trial, A_n), color = "#088880", size = 10^(-10))

ggplot(Data_An) + geom_point(aes(Trial, A_n), color = "#088880", size = 10^(-10)) + transition_manual(Trial)

For, n = 1000, we plot the gaps instead of the points of \(A_n\) itself.

A = c(0)
K = 3
Data_Gap = data.frame(Trial = c(0), Gaps = c(0))
for (i in 1:1000) {
  A = A_n(A, K)
  Gaps = setdiff(seq(min(A),max(A), by = 1),A)
  for (p in 1: length(Gaps)){
    Data_Gap = Data_Gap %>% add_row(Trial=i, Gaps = Gaps[p])
  }
}

ggplot( Data_Gap ) + geom_point(aes(Trial,Gaps), size =1, color = "#088880") + transition_manual(Trial, cumulative = T) +  enter_fade()

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A = c(0)
K = 3
trial = seq(1,1000,by = 1)
G2_An = c()
L_An = c()
for (i in trial) 
{
  A = A_n(A,K)
  gaplength = diff(sort(A)) - 1
  G2_An[i] = sum(gaplength >=2)
  L_An[i] = sum(gaplength >=1)
}

A = c(0)
K = 3
trial = seq(1,1000,by = 1)
for (h in 2:100) 
{
  for (i in trial) 
  {
    A = A_n(A,K)
    gaplength = diff(sort(A)) - 1
    G2_An[i] = G2_An[i] + sum(gaplength >=2)
    L_An[i] =  L_An[i] + sum(gaplength >=1)
  }
}
Gn = G2_An / 100
Ln = L_An / 100

Using the given theorem 2.1.1 in Dorothea Marie Wagner - 1999, \[ lim \text{ } sup_{n \rightarrow \infty} \frac{L(A_n)}{h(n)} = \frac{K-1}{k} = \frac{2}{3}\] \[ \implies \overline{\frac{L(A_n)}{h(n)}} \geq \frac{2}{3}\] \[ \implies \exists N, \text{ such that }L_n = mean(L(A_n)) \leq \overline{L(A_n)} \leq \frac{2}{3}h(n) \text{ for all } n \geq N\]

plot(seq(1,1000, by = 1),Gn, type = "l",col = "#088880",  main = "Plot of averages of no.of gaps with length atleast 2 over values of n", xlab = "n values", ylab = "Gn")

(https://arxiv.org/pdf/math/9908030.pdf,)

Apple Bread Challenge Problem 2

Samadrita Bhattacharya