Exponent-pair-type bound for exponential sums of square roots

Establish whether, for all integers n ≥ 1 and l ≠ 0, the exponential sum S(n,l) = ∑_{a=1}^{n} exp(2π i l √a) satisfies a bound of the form |S(n,l)| ≤ C_ε · l^ε · n^{1/2}, uniformly in n and l, for every ε > 0. Determine if this exponent-pair-type estimate holds for the explicit phase √a and quantify its validity or failure.

Background

The paper studies how closely sums of square roots of integers can approach an integer and uses a Fourier-analytic method that reduces the problem to controlling exponential sums of the form ∑_{a=1}^{n} e^{2π i l √a}. The main bottleneck in the argument is obtaining sufficiently strong bounds for these exponential sums to guarantee many nontrivial solutions near any point mod 1.

The authors note that the exponent pair hypothesis would suggest an estimate of size l^ε n^{1/2} for the sum, which, if true, would yield substantially stronger proximity results (on the order of n^{-k/2+ε}) for sums of k square roots. They explicitly state uncertainty about whether such a bound can be proved for this explicit phase, highlighting it as an unresolved question critical to improving the main results.