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In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], several inequalities for tail probabilities of sums Mn = X1 + · · · + Xn of bounded independent random variables Xj were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. HautesÉtudes Sci. Publ. Math. 81 (1995a) 73-205] inserted certain missing factors in the bounds of two theorems. By similar factors, a third theorem was refined by Pinelis [Progress in Probability 43 (1998) 257-314] and refined (and extended) by me. In this article, I introduce a new type of inequality. Namely, I show that P{Mn ≥ x} ≤ cP{Sn ≥ x}, where c is an absolute constant and Sn = ε1 + · · · + εn is a sum of independent identically distributed Bernoulli random variables (a random variable is called Bernoulli if it assumes at most two values). The inequality holds for those x ∈ R where the survival function x → P{Sn ≥ x} has a jump down. For the remaining x the inequality still holds provided that the function between the adjacent jump points is interpolated linearly or log-linearly. If it is necessary, to estimate P{Sn ≥ x} special bounds can be used for binomial probabilities. The results extend to martingales with bounded differences. It is apparent that Theorem 1.1 of this article is the most important. The inequalities have applications to measure concentration, leading to results of the type where, up to an absolute constant, the measure concentration is dominated by the concentration in a simplest appropriate model, such results will be considered elsewhere.

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For d-dimensional irrational ellipsoids E with d ≥ 9 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order o(r d−2 ). The estimate refines an earlier authors' bound of order O(r d−2 ) which holds for arbitrary ellipsoids, and is optimal for rational ellipsoids. As an application we prove a conjecture of Davenport and Lewis that the gaps between successive values, say s < n(s),

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