The ‘Three Squares’ Problem and Related Questions

Abstract

Let n ⩾ 1 and let a ₁,... , a _n+1 be fixed positive numbers. We consider the following problem. Let f ∊ L _loc(ℝⁿ) and assume that

$$ \int\limits_{\left[ { - a_j ,a_j } \right]n} {f\left( {x + y} \right)dx = 0} $$

(6.1)

for all y ∊ ℝⁿ and j = 1,...,n+ 1. For what numbers a ₁,...,a _n+1 does this imply that f = 0? If n = 2 then we have the so called ‘three squares’ problem. This problem and its generalizations have been studied by many authors (see [B14], [B29], [L2], [V3], [V9], [V33], [S11], [S12]). In particular, they have shown that the equality (6.1) implies that f = 0 if and only if every ratio a _i /a _j (1 ⩽ i, j ⩽ n + 1, i ≠ j) is irrational. This means that Fourier transforms for indicators of cubes [−a _j ,a _j ]ⁿ, 1 ⩽ j ⩽ n + 1 have no common zeroes. It is easy to see that for any (fixed) a ₁,...,a _n > 0 there exists a nonzero function f ∊ C ^∞(ℝⁿ) satisfying (6.1) for all y ∊ ℝⁿ, j = 1,. .., n. For example, the function f(x) = e ^i(x,u) has a such property for some u ∊ ℝⁿ.