input: 
  
  def find_matches(query: str, choices: List[str], threshold: int) -> List[str]:
      """
      This function performs a fuzzy search on the given list of choices to find those 
      that are similar to the query string based on a specified threshold for 
      Levenshtein distance. It converts all strings to lower case before comparison 
      and returns a list of choices that match the query within the distance threshold.
      
      >>> find_matches('apple', ['apples', 'banana', 'app'], 2)
      ['apples', 'app']
      
      >>> find_matches('book', ['back', 'books', 'boo', 'boost'], 1)
      ['back', 'boo']
      """
      query, choices = query.lower(), [choice.lower() for choice in choices]
      matches = []
      
      for choice in choices:
          distance = levenshtein_distance(query, choice)
          if distance <= threshold:
              matches.append(choice)
      
      return matches
  
output:     from typing import List
  
      def levenshtein_distance(s1: str, s2: str) -> int:
          if len(s1) < len(s2):
              return levenshtein_distance(s2, s1)
  
          if len(s2) == 0:
              return len(s1)
  
          previous_row = range(len(s2) + 1)
          for i, c1 in enumerate(s1):
              current_row = [i + 1]
              for j, c2 in enumerate(s2):
                  insertions = previous_row[j + 1] + 1
                  deletions = current_row[j] + 1
                  substitutions = previous_row[j] + (c1 != c2)
                  current_row.append(min(insertions, deletions, substitutions))
              previous_row = current_row
  
          return previous_row[-1]