The ‘Three Squares’ Problem and Related Questions

V. V. Volchkov
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V. V. Volchkov: Donetsk National University, Department of Mathematics

Chapter Chapter 6 in Integral Geometry and Convolution Equations, 2003, pp 311-319 from Springer

Abstract: Abstract Let n ⩾ 1 and let a 1,... , a n+1 be fixed positive numbers. We consider the following problem. Let f ∊ L loc(ℝ n ) and assume that 6.1 $$ \int\limits_{\left[ { - a_j ,a_j } \right]n} {f\left( {x + y} \right)dx = 0} $$ for all y ∊ ℝ n 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 ∞(ℝ n ) satisfying (6.1) for all y ∊ ℝ n , j = 1,. .., n. For example, the function f(x) = e i(x,u) has a such property for some u ∊ ℝ n .

Keywords: Inductive Hypothesis; Direct Calculation; Local Version; Relate Question; Fourier Trans (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-010-0023-9_24

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DOI: 10.1007/978-94-010-0023-9_24

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