Sharp Results Concerning the Expression of Functions as Sums of Finite Differences

Let * denote convolution and let δ x denote the Dirac measure at a point x . A function in L 2 ( R )) is called a difference of order 1 if it is of the form g ‐δ x * g for some x ∈ R and g ∈ L 2 ( R )). Also, a difference of order 2 is a function of the form g − 2 − 1 ( δ x ∗ g + δ − x ∗ g ) for some x ∈ R and g ∈ L 2 ( R )). In fact, the concept of a ‘difference of order s ’ may be defined in a similar manner for each s 0. If f denotes the Fourier transform of f , it is known that a function f in L 2 ( R )) is a finite sum of differences of order s if and only if ∫ − ∞ ∞|f ^ ( x ) | 2 | x | − 2 s d x < ∞ , and the vector space of all such functions is denoted by D s ( L 2 ( R )). Every function in D s ( L 2 ( R )) is a sum of int(2 s ) + 1 differences of order s , where int( t ) denotes the integer part of t . Thus, every function in D 1 ( L 2 ( R )) is a sum of three first order differences, but it was proved in 1994 that there is a function in D 1 ( L ( R )) which is never the sum of two first order differences. This complemented, for the group R , the corresponding result for first order differences obtained by Meisters and Schmidt in 1972 for the circle group. The results show that there is a function in L 2 R such that, for each s ⩾ 1/2, this function is a sum of int (2 s ) + 1 differences of order s but it is never the sum of int (2 s ) differences of order s . The proof depends upon extending to higher dimensions the following result in two dimensions obtained by Schmidt in 1972 in connection with Heilbronn's problem: if x 1 , x _ n are points in the unit square,∑ 1 ⩽ i < j ⩽ n| x i − x j| − 2 ⩾ 200 − 1n 2 ln ⁡ n . Following on from the work of Meisters and Schmidt, this work further develops a connection between certain estimates in combinatorial geometry and some questions of sharpness in harmonic analysis. 2000 Mathematics Subject Classification 42A38 (primary), 52A40 (secondary).