关于和集与谱隙
On Sumsets and Spectral Gaps
摘要：
众所周知，如果S是整数mod p的子集，并且如果第二大的傅里叶系数相对于最大的系数是"小"的，则和集S + S比S大得多。我们在本文中表明，如果在最大和第二大傅里叶系数之间没有如此大的"谱隙"，而是在第k大和第（k +1）大之间，同样的事情也成立，即|S + s|明显大于|S|。我们只在k <（log p）/（log 4）的情况下这样做。对于重复和集S + S +.+，我们也得到了类似的结果。S，结果是包含的项越多，可以使用的索引k就越大。
英文摘要：
It is well known that if S is a subset of the integers mod p, and if the second-largest Fourier coefficient is ``small'' relative to the largest coefficient, then the sumset S+S is much larger than S. We show in the present paper that if instead of having such a large ``spectral gap'' between the largest and second-largest Fourier coefficients, we had it between the kth largest and the (k+1)st largest, the same thing holds true, namely that |S+S| is appreciably larger than |S|. Well, we only do this for k < (log p)/(log 4). We also obtain analogous results for repeated sumsets S+S+...+S, and it turns out that the more terms one includes, the larger the index k that can be used.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
