Research on density estimates for exceptional sets associated with additive representations of natural numbers

这项为期三年的研究项目最核心的成就是，我们成功地完全确定了不能写成最多16个双二次和的整数。我们特别证明了，通过使用圆法，每个超过10的216次方且不能被16整除的整数都可以写成16个双二次方的和。我们还明确地确定了与17个和18个双二次方之和相关的例外集。例如，我们确定每个自然数都可以写成18个双二次方的和，但只有以下7个数字除外：79、159、239、319、399、479和559。此外，作为副产品，我们还给出了一个新的证明，证明了每个自然数最多可以写成19个双二次方的和。其次，关于Wering-Goldbach问题，我们证明了每个足够大的整数都可以写成七个自然数立方体的和，这些立方体的自然数最多有四个按重数计算的素因数。这一结论改进了以前在这个方向上的最好结果，在这个方向上，素因数的数量上限是69个而不是4个。后一个定理是用筛法和圆法证明的，除了上面提到的两个主要乘积外，我们还研究了与下列加法问题有关的例外集：7个光滑数立方体之和(即只有较小素数因子的自然数)之和，15个双二次体之和，以及5个质数立方体之和。我们打算继续在这一领域进行研究。

The most central accomplishment of this three year research project is that we succeeded in determining completely the integers that cannot be written as the sum of at most sixteen biquadrates. We in particular contributed to the proof that every integer exceeding 216th power of 10 and not divisible by 16 can be written as the sum of sixteen biquadrates, by making use of the circle method. We also determined explicitly the exceptional sets associated with the sum of seventeen and eighteen biquadrates. For example, we established that every natural number can be written as the sum of eighteen biquadrates except only for the following seven numbers; 79, 159, 239, 319, 399, 479 and 559. As a by-product, moreover, we provided a new proof, substantially simpler than the previous one, of the fact that every natural number can be written as the sum of at most nineteen biquadrates.Next, concerning the Waring-Goldbach problem, we showed that every sufficiently large integer can be written as the sum of seven cubes of natural numbers that have at most four prime factors counted according to multiplicity. This conclusion improves the previously best result in this direction, in which the upper limit for the number of prime factors was 69 in place of four. The latter theorem was proved by applying sieve methods and the circle method.Besides these two main products mentioned above, we worked on the exceptional sets associated with the following additive problems; sums of seven cubes of smooth numbers (that means natural numbers having smaller primes factors only), sums of fifteen biquadrates, and sums of five cubes of primes. We intend to continue our research in this area.