Abstract
Improving a result of Hajela, we show for every function f with limn→∞f(n) = ∞ and f(n) = o(n) that there exists n0 = n0(f) such that for every n ⩾ n0 and any S ⊆ {–1, 1}n with cardinality |S| ⩽ 2n/f(n) one can find orthonormal vectors x1, …, xn ∈ ℝn satisfying
Funding statement: The research of the first named author is supported by the National Scholarship Foundation (IKY), sponsored by the act “Scholarship grants for second-degree graduate studies”, from resources of the operational program “Manpower Development, Education and Life-long Learning”, 2014-2020, co-funded by the European Social Fund (ESF) and the Greek state. The second named author is supported by a PhD Scholarship from the Hellenic Foundation for Research and Innovation(ELIDEK); research number 70/3/14547.
Communicated by: M. Henk
Acknowledgements
We would like to thank Apostolos Giannopoulos for useful discussions and the anonymous referees for their comments and valuable suggestions that helped to improve the presentation of the results of this article.
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