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The well-known k-SUM conjecture is that integer k-SUM requires time Omega(n^{\ceil{k/2}-o(1)}). Recent work has studied multidimensional k-SUM in F_p^d, where the best known algorithm takes time \tilde O(n^{\ceil{k/2}}). Bhattacharyya et al. [ICS 2011] proved a min(2^{\Omega(d)},n^{\Omega(k)}) lower bound for k-SUM in F_p^d under the Exponential Time Hypothesis. We give a more refined lower bound under the standard k-SUM conjecture: for sufficiently large p, k-SUM in F_p^d requires time Omega(n^{k/2-o(1)}) if k is even, and Omega(n^{\ceil{k/2}-2k(log k)/(log p)-o(1)}) if k is odd. For a special case of the multidimensional problem, bounded monotone d-dimensional 3SUM, Chan and Lewenstein [STOC 2015] gave a surprising \tilde O(n^{2-2/(d+13)}) algorithm using additive combinatorics. We show this algorithm is essentially optimal. To be more precise, bounded monotone d-dimensional 3SUM requires time Omega(n^{2-\frac{4}{d}-o(1)}) under the standard 3SUM conjecture, and time Omega(n^{2-\frac{2}{d}-o(1)}) under the so-called strong 3SUM conjecture. Thus, even though one might hope to further exploit the structural advantage of monotonicity, no substantial improvements beyond those obtained by Chan and Lewenstein are possible for bounded monotone d-dimensional 3SUM.

Homogeneous isotropic multi-variable random function is used to describe the fluctuation of the 3-dimensional density discontinuity surface. A relationship between the gravity anomaly and the change of the discontinuity surface buried depth is derived on the basis of the statistical equivalent method. The result is used to analyze the relation of linear correlation between the crustal thickness and the local Bouguer gravity anomaly, and good agreement is obtained in comparison with the actual statistical data. (Author), Pu kuei tse san wei mi tu fen chieh mien wen t'i, trans. of Ti Ch'iu Wu Li Hsueh Pao (Chinese People's Republic) v24 n1 p56-65 1965.