Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums

We prove non-trivial bounds for bilinear forms with hyper-Kloosterman sums with characters modulo a prime $q$ which, for both variables of length $M$, are non-trivial as soon as $M\geq q^{3/8+\delta}$ for any $\delta>0$. This range, which matches Burgess's range, is identical with the best results previously known only for simpler exponentials of monomials. The proof combines refinements of the analytic tools from our previous paper and new geometric methods. The key geometric idea is a comparison statement that shows that even when the "sum-product" sheaves that appear in the analysis fail to be irreducible, their decomposition reflects that of the "input" sheaves, except for parameters in a high-codimension subset. This property is proved by a subtle interplay between \'etale cohomology in its algebraic and diophantine incarnations. We prove a first application concerning the first moment of a family of $L$-functions of degree $3$.Comment: 58 pages; minor cosmetic corrections; to appear in Annali della Scuola Normale Superiore di Pis

Risk factors for low urinary citrate in calcium nephrolithiasis: low vegetable fibre intake and low urine volume to be added to the list

Risk factors for low urinary citrate excretion were assessed in 34 consecutive male recurrent idiopathic calcium stone formers (RCSF) who collected two 24-h urines while on free-choice diet. Overt hypocitraturia (hypo-cit) was defined as UCit×V<1.70 mmol/day, and ‘low' citraturia (low-cit) as UCit×V between 1.70 and 2.11 mmol/day. Twenty-three RCSF had normocitraturia (normo-cit), six low-cit and five hypo-cit. UCit×V positively correlated with urine volume (VOLUME, r=0.44, P=0.009), vegetable fibre intake (fibers, r=0.46, P=0.009) and GI-alkali absorption (alkali, r=0.47, P=0.006), and volume, fibres and alkali tended to be lower among RCSF with low-/hypo-cit. A 3-day NH4Cl loading test (0.95 mEq/kg BW daily in 3 doses) was performed in RCSF as well as in 14 age-matched healthy male controls (C). On a plot of urine pH versus serum bicarbonate, 10 of 11 RCSF with low-/hypo-cit, but only six of 23 with normo-cit (P=0.0004) fell off the normal range, indicating incomplete RTA. Two or more risk factors simultaneously occurred in only four of 23 RCSF with normo-cit, but in eight of 11 with low-/hypo-cit (P= 0.002). In conclusion, incomplete RTA is the most prevalent risk factor for low-/hypo-cit in RCSF, and decreases in vegetable fibres and urine volume emerge as two new risk factors for low urinary CI