2020/11/11 by Reynold Fregoli · 1 citation
Mathematics · #Mathematical Dynamics and Fractals #Functional Equations Stability Results #Advanced Combinatorial Mathematics
paper · doi:10.1112/blms.12431
Let A ⊂ N , α ∈ ( 0 , 1 ) , and e ( x ) : = e 2 π i x for x ∈ R . We set S A ( α , N ) : = ∑ n ∈ A n ⩽ N e ( n α ) . Recently, A'Campo posed the following question: Is there an infinite non-cofinite set A ⊂ N such that for all α ∈ ( 0 , 1 ) the sum S A ( α , N ) has bounded modulus as N → + ∞ ? In this note, we show that such sets do not exist. To do so, we use a theorem by Duffin and Schaeffer on complex power series. We extend our result by proving that if the sum S A ( α , N ) is bounded in modulus on an arbitrarily small interval and on the set of rational points, then the set A has to be either finite or cofinite. On the other hand, we show that there are infinite non-cofinite sets A ⊂ N such that | S A ( α , N ) | is bounded independently of N for all α ∈ E ⊂ ( 0 , 1 ) , where Q ∩ ( 0 , 1 ) ⊂ E and E has full Hausdorff dimension.