## Furstenberg–Sárközy theorem |

The **Furstenberg–Sárközy theorem** is a result in
^{
[1]}

An example of a set with no square differences arises in the game of
^{
[2]}

Any position in this game can be described by an integer, its number of coins. The non-negative integers can be partitioned into "cold" positions, in which the player who is about to move is losing, and "hot" positions, in which the player who is about to move can win by moving to a cold position. No two cold positions can differ by a square, because if they did then a player faced with the larger of the two positions could move to the smaller position and win. Thus, the cold positions form a set with no square difference:

These positions can be generated by a
^{
[2]}^{
[3]} As Golomb observed, the cold positions are infinite, and more strongly the number of cold positions up to is at least proportional to . For, if there were fewer cold positions, there wouldn't be enough of them to supply a winning move to each hot position.^{
[2]} The Furstenberg–Sárközy theorem shows, however, that the cold positions are less frequent than hot positions: for every , and for all large enough , the proportion of cold positions up to is at most . That is, when faced with a starting position in the range from 1 to , the first player can win from most of these positions.^{
[4]}