On the signed small-ball inequality with restricted coefficients (Q2822154)

Let \(R=R_1\times R_2\times \cdots \times R_d\) be a dyadic rectangle in the unit cube \([0,1]^d\). Let \(h_R\) denote the \(L^{\infty}\) normalized Haar function supported on \(R\). The small ball conjecture is as follows: For \(d\geq 2\), coefficients \(\alpha_R\), and an integer \(n\geq 1\), there exists a positive constant \(C_d\) such that NEWLINE\[NEWLINE n^{(d-2)/2}\|\sum_{|R|=2^{-n}}\alpha_R h_R\|_{\infty} \geq C_d\cdot 2^{-n}\sum_{|R|=2^{-n}} |\alpha_R|.NEWLINE\]NEWLINE In this paper the author proofs that for \(d\geq 3\), an integer \(n\geq 1\), and coefficients \(\alpha_R\in \{-1, 1\}\) with the splitting property (\(\alpha_R=\alpha_{R_1}\alpha_{R'}\)), there exists a positive constant \(C_d\) such that NEWLINE\[NEWLINE\|\sum_{|R|=2^{-n}}\alpha_R h_R\|_{\infty} \geq C_d\cdot n^{d/2}.NEWLINE\]